UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


ELEMENTS 


NATURAL  PHILOSOPHY, 

BY  E.  S. /FISCHER, 

HONORARY   MEMBER    OF   THE    ACADEMY    OF    SCIENCES    OF    BERLIN, 

PROFESSOR    OF   MATHEMATICS    AND    N\ATURAL    PHILOSOPHY    IN    ONE    OF   THE 

COLLEGES    OF   THE    SAME    CITY,    &.C.    &.C. 

TRANSLATED  INTO  FRENCH 
WITH 

NOTES  AND  ADDITIONS, 

BY  M.  BIOT, 

OF   THE '•INSTITUTE    OF    FRANCE, 
AND   NOW 

TRANSLATED  FROM  THE  FRENCH  INTO  ENGLISH 

FOR   THE 

USE  OF  COLLEGES  AND  SCHOOLS 

IN   THE 

UNITED   STATES. 


EDITED 

BY    JOHN   FARRAR, 

OK    MATHEMATICS    AND    NATURAL    PHILOSOPHY    IN   THE 
UNIVERSITY    AT    CAMBRIDGE,    NEW    ENGLAND. 


BOSTON  : 
HILLIARD,  GRAY,  LITTLE,  AND  WILKINS. 

1827. 


DISTRICT  OF  MASSACHUSETTS,  TO  WIT. 

District  Clerk's  Office. 

BE  it  remembered,  that  on  the  twenty-fourth  day  of  November,  1827,  in  the 
fifty-second  year  of  the  Independence  of  the  United  States  of  America,  Hil- 
liard,  Gray,  &  Co.,  of  the  said  district,  have  deposited  in  this  office  the  title 
of  a  book,  the  fight  whereof  they  claim  as  proprietors,  in  the.  words  fol- 
lowing, viz : 

"  Elements  of  Natural  Philosophy,  by  E.  S.  Fischer,  honorary  Member  of 
the  Academy  of  Sciences  of  Berlin,  Professor  of  Mathematics  and  Natural 
Philosophy  in  one  of  the  Colleges  of  the  same  city,  &.c  &c.  Translated  into 
French,  with  Notes  and  Additions,  by  M.  Biot,  of  the  Institute  of  France  ; 
and  now  translated  from  the  French  into  English  for  the  use  of  Colleges  and 
Schools  in  the  United  States.  Edited  by  John  Farrar,  Professor  of  Mathe- 
matics and  Natural  Philosophy  in  the  University  at  Cambridge,  New  Eng- 
land." 

In  conformity  to  the  act  of  the  Congress  of  the  United  States,  entitled  "  An 
act  for  the  encouragement  of  learning,  by  securing  the  copies  of  maps, 
charts,  and  books,  to  the  authors  and  proprietors  of  such  copies,  during  the 
times  therein  mentioned  ;"  and  also  to  an  act,  entitled,  •'  An  act  supplementa- 
ry to  an  act,  entilled,  '  An  act  for  the  encouragement  of  learning,  by  securing 
the  copies  of  maps,  charts,  and  books,  to  the  authors  and  proprietors  of  such 
copies,  during  the  times  therein  mentioned,'  and  extending  the  benefits 
thereof  to  the  arts  of  designing,  engraving,  and  etching,  historical  and  other 
prints." 

JNO.    W.  DAVIS, 
Clerk  of  the  District  of  Massachusetts 


CAMBRIDGE. 

Hilliard,  Metcalf,  and  Company 
Printers  to  the  University. 


CD  C  <3  \ 

V53 


ADVERTISEMENT. 


THE  comprehensive  and  elementary  outline  of  Natural 
Philosophy  by  E.  S.  Fischer,  of  which  the  following  is  a 
translation,  has  been  much  used  and  highly  approved  in 
the  German  and  French  schools.  It  was  selected  by 
Biot,  himself  the  author  of  two  very  valuable  works  upon 
the  same  subject,  as  best  suited  to  a  certain  class  of  stu- 
dents, whose  want  of  leisure,  or  of  previous  preparation, 
might  preclude  them  from  more  ample  and  elaborate 
treatises.  A  few  alterations  and  additions  have  been 
made  for  the  purpose  of  adapting  it  to  the  use  of  learners 
in  the  public  and  private  seminaries  of  the  United 
States.  The  alterations  consist  principally  in  converting 
the  French  weights,  measures,  degrees,  &c.  into  the  cor- 
responding English  denominations.  The  additions  are 
distinguished  by  being  in  brackets. 

Cambridge,  November,  1827. 


. 


4C4526 


CONTENTS. 


SECTION  I. 

OF  BODIES  IN  GENERAL. 

CHAP.  I. — General  Considerations  respecting  the  Properties 

which  belong  to  all  Bodies        ......  1 

CHAP.  II. — State  of  Aggregation  of  Bodies         ...  3 

CHAP.  III. — Great  Variety  of  Material  Properties  in  Bodies  5 

CHAP.  IV. — Different  Modes  of  Considering  Bodies         .         .  8 

Atomic  System     ......  ib. 

Dynamic  System ib. 

Empirical  Considerations      ....  9 

CHAP.  V. — Mathematical  Laws  of  Motion    .         .         .         .10 

CHAP.  VI. — Physical  Laws  of  Motion,  or  Investigation  of  Mov- 
ing Forces      ........  12 

CHAP.  VII. — Historical    View  of  what  is  known  respecting 

Gravity 15 

SECTION  II. 

SOLID  BODIES. 

CHAP.  VIII. — General  Properties  of  Solid  Bodies          .         .18 

CHAP.  IX.— The  Interior  Construction  of  Solid  Bodies       .  21 
CHAP.  X. — Equilibrium  of  Solid  Bodies,   or  Fundamental 

Principles  of  Statics 23 

Equilibrium  of  Free  Bodies  ib. 
Equilibrium  of  Bodies  which  move  about  a  Fixed 

Axis 24 

CHAP.  XI.— Centre  of  Gravity  of  Bodies          ...  25 
CHAP.  XII. — Free  Descent  of  Heavy  Bodies. — Laws  of  Mo- 
tion uniformly  accelerated    .          .....  27 

Motion  down  Inclined  Planes          ...  32 

CHAP.  XIII.— Free  Curvilinear  Motion        ....  34 

(1.)    Projectile^  \      T  ib. 

(2.)    Central  Forces 36 


vi  Contents. 

CHAP.  XIV. — Motion  in  given  Lines        ....  37 

(1.)   Curvilinear  Motions           .         .         .  .     ib. 

(2.)   Oscillations  of  the  Pendulum            .         .  38 

(3.)  Application  of  the  Pendulum       .         .  .42 

CHAP.  XV. —  Communication  of  Motion  by  Impulse        .  .     47 

CHAP.  XVI. — Vibratory  Motions,  and  the  Sounds  which  they 

produce;  or  the  first  Principles  of  Acoustics  .         .  .52 

SECTION  HI. 

HEAT. 
CHAP.  XVII. — Heat  in  general ;  its  expansive  Force  ;   T/ier- 

momeltr  and  Pyrometer    ......         57 

Addition "67 

CHAP.  XVIII. — Changes  produced  by  Heat  in  the  State  of 

Aggregation  of  Bodies     ......         GS 

CHAP.  XIX. — The  Propagation  of  Heat      .         .         .         .73 

The  Calorimeter 75 

CHAP.  XX. — Production  of  Heat  and  Cold          .         .         .79 
Artificial  Cold 81 

SECTION  IV. 

LIQUID  BODIES. 
CHAP.  XXI. — Liquids  in  General       .         .         .         .         .83 

Water ft. 

Mercury 85 

Alcohol        .....  86 

Ether '    .     '    .  87 

General  Remarks  upon  Liquids      .         .         .  ib. 

CHAP.  XXII.— The  Specific  Gravity  of  Solids  and  Liquids  88 
CHAP.  XXIII.— Equilibrium  of  Liquids,  or  First  Principles  of 

Hydrostatics     ....'..  94 
Pressure  of  a  Liquid  against  the  Bottom  and  Sides 

of  a  Vessel gg 

Pressure  of  a  Liquid  upon  Solid  Bodies  immersed 

init 97 

Floating  Bodies gg 

CHAP.  XXIV. — Hydrostatic  Balance  and  Hydrometer  .         .  99 
Areometer  or  Hydrometer       .         .         .  IQI 
CHAP.  XXV.     Influence  of  Adhesion  and  Cohesion  upon  Hy- 
drostatic Phenomena          ...  103 


Contents.  vii 

CHAP.  XXVI. — Motions  of  Liquids,  or  First  Principles    o 

Hydraulics  .         .         .         .    •    .         .         .         Ill 

Hydraulic  Experiments  confirming  the  preceding 

Theory 114 

Influence  of  Forces  different  from  Gravity  upon 

Hydraulic  Motions       .         .         .         .         .       ib. 
The  Motions  of  Solid  Bodies  in  Liquids        .         117 

SECTION  V. 

AERIFORM  BODIES. 

CHAP.  XXVII.— Elastic  Fluids  in  General        .         .         .119 
Atmospheric  Air  .....  ib. 

Oxygen  ,  -  . •»*>-.••*:«.  .  .&•;.  .  .  121 
Azote  .  .  »V-  *>...*.  .  122 
Hydrogen  .  >-.,-..  .•%*-» x  ;  jfi •  .  .  123 
Carbonic  Acid  Gas  .  .  .  <£ ''  .  .  ib. 

Elastic  Vapours 124 

CHAP.  XXVIII. —  Water  in  Atmospheric  Air,  or  First  Princi- 
ples of  Hygrometry         .         .          .         .          .          .          125 

Addition  relative  to  Hygrometry  .  .  .128 
General  Remarks  upon  Hygrometry  .  .  133 

Addition 134 

CHAP.  XXIX. — Barometer  and  Air-Pump     .         .     •  /  *        137 
Barometer     .         ...         .         .         .         .       ib. 

Air-Pump  .         .         .         .         .         .          141 

Condensing  Pump  .         .         .         .         .143 

Mechfinical  Properties  of  Air         .         .         .         144 

DUatability  of  the  Air ib. 

Pressure  of  the  Air 146 

Gravity  of  the  Air  .         .         .         .         .148 

Specific  Gravity  of  other  Gases     .         .         .         149 
CHAP.  XXX.— Equilibrium  of  Air,  or  First  Principles  of  Ae- 
rostatics .         .         .         .         .         .         .         .  ib. 

Law  of  Mariotte  ;  or  Ratio  of  the  Pressure  and 

Elasticity  to  the  Density  or  Specific  Gravity         150 
Law  by  which  the  Density  of  the  Air  decreases  as 

we  ascend  into  the  Atmosphere        .         .         .153 
More  exact  Estimate  of  the  Influence  which  Heat 
lias  upon  the  Mechanical  Properties  of  a  Dilata- 
ble Fluid   155 


vjjj  Contents. 

Measurement  of  Heights  by  the  Barometer          .     1 50 
Height  of  the  Atmosphere     .  * 

CHAP.  XXXI.— Motions  of  Elastic  Fluids          .         .         .     150 

SECTION  VI. 

ELECTRICITY. 
CHAP.  XXXIL— Electrical  Machine,  and  General  Phenomena 

of  Electricity 

CHAP.  XXXIIL— Opposite  Electricities      ....     171 
Electrical  Phenomena  in  the  Dark  and  in  Rarefied 

Air 173 

Hypothesis  of  Franklin I"7  5 

Hypothesis  of  Symmer 

Addition.— The  Electric  Balance      .         .         .177 
CHAP.  XXXIV.— Striking  Distance,  Sphere  of  Activity,  Ac- 
cumulated Electricity 179 

Sphere  of  Activity 180 

Accumulated  Electricity 184 

CHAP.  XXXV. — Electrophorus  and  Condenser        .         .         190 
Electrophorus          .         .         .         •         •         .       ib. 

Condenser 193 

CHAP.  XXXVI.— Electricity  excited  by  other  Means  beside 

Friction 194 

Galvanism          .         .         .         .         .         .         196 

Voltaic  Pile,  or  Galvanic  Battery    .         .         .198 
Relations  of  Electricity  and  Galvanism  .        201 

Addition       .......       ib. 

SECTION  VII. 

MAGNETISM. 

CHAP.  XXXVII.— General  Properties  of  the  Magnet        .         211 
Relation  between  the  Magnet  and   Unmagnetized 
Iron      -c''.    •.';,...    r.    •     .    .     .         .     212 

Properties  of  the  Magnet      .         .         .         .         213 

The  Reciprocal  Action  of  Magnets    .         .         .214 
Communication  of  Magnetism        .         / ;      .  ib. 

Distribution  of  Magnetism,  and  Sphere  of  Mag- 
netic Activity       .         .         .         .         .          .215 

CHAP.  XXXVHI.— More  Particular  Examination  of  the  Phe- 
nomena of  the  Magnetized  Netdle        .         .         .         .217 


Contents.  ix 

Declination  Needle 217 

Dipping  Needle 219 

Terrestrial  Magnetism          .         .         .         .  221 

Excitation  of  Natural  Magnetism     .         .         .  222 

Addition    .......  ib. 

SECTION  VIII. 

OPTICS. 

CHAP.  XXXIX. — Of  Light  in  General;  particularly  the  Phe- 
nomena which  depend  upon  its  Motion  in  a  Right  Lane  ; 

or  First  Principles  of  Optus 225 

Mechanical  Phenomena  of  Direct  Light     .         .  227 

CHAP.  XL.— Vision 233 

CHAP.  XLI. — Reflection  of  Light  by  Mirrors,  or  First  Princi- 
ples of  Catoptrics            .       -.-.'.'        .   £>.         .  239 
Fundamental  Law  of  Catoptrics    ,     F.                .  240 
Plane  Mirrors    .         .         .  ,    ,  ,  £ .         .  241 
Spherical  Mirrors  ......  242 

Phenomena  produced  by  Concave  Mirrors       .  ib. 

Phenomena  produced  by  Convex  Mirrors            .  245 

Mathematical  Additions        ....  247 

CHAP.  XLII. — Refraction  of  Light  in  Transparent  Bodies  ;  or 

First  Principles  of  Dioptrics             ....  253 

Law  cf  Dioptrics    ......  254 

General  Phenomena  which  depend  upon  the  Re- 
fraction of  Light         .          .         .         .          .257 

Particular  Phenomena  which  are  produced  by  means 

of  Polished  Glasses           .         .         .         .  258 

Plane  Glasses  ivith  Parallel  Faces     .         .         .  ib. 

Spherical  Glasses  or  Lenses           .         .         .  259 

Phenomena  produced  by  Converging  Glasses  261 

Phenomena  produced  by  Diverging  Glasses         .  267 

Mathematical  Additions        ....  268 

CHAP.  XLIII. —  Compound  Optical  Instruments  .         .         .  275 

A.  Refracting  Telescope       ....  ib. 

Phenomena  produced  by   means   of    Converging 

Glasses  when  the  Object  is  behind  the  Glass  277 
Phenomena   produced    by    means    of  Diverging 

Glasses,  when  the  Object  is  behind  the  Glass    .  27ft 


Contents. 

B.  The  most  Important  Kinds  of  Refracting  Tele- 
scopes        .......     278* 

C.  Compound  Microscope    .         .         .         .         281 
CHAP.  XL1V. — Theory  of  Dioptric  Colours,  or  the  Decompo- 
sition of  Light         V    .   *    '    .       .   «*;<.<••    .        .    283 

The  Glass  Prism  .  .  •'..•'  •  ib- 

Colours  produced  by  Thin  Lamince  .  .  289 

General  Observations  upon  Newton's  Theory  of 

Colours 290 

Effects  of  the  Dispersion  of  Colours  in  Optical 

Glasses 291 

CHAP.  XLV. — Reflecting  Telescope  and  Achromatic  Lenses  292 
State  of  Optics  before  Newton's  Time  .,  .  ib. 

Errors  of  Newton  in  the  Theory  of  Colours  .  293 
Reflecting  Telescope  ....  ••'  .  294 
Ingenious  Researches  of  Euler ;  his  Errors. — 

Dollond. — Klingenstitrn  '  .  . 9  .  296 
Mathematical  Additions  .  .  .  .  j'jo 

APPENDIX  TO  OPTICS. 

Coloured  Rings 306 

Another  Explanation  of  the  Coloured  Rings  on  the  Hypothe- 
sis of  Undulations.— Dr  Young's  Principle  of  Interferences  314 

Diffraction  of  Light       '   .'       .         .        ..  •      .         .  321 

Double  Refraction         .         .         .         .         .         .- v    ''.         324 

Polarisation  of  Light        .         .         .         .         .•<•'•.•'          336 


SECTION  I. 


OF   BODIES   IN   GENERAL 


CHAPTER  I. 

General  Considerations  respecting  the  Properties  which  belong  to  alt 
Bodies. 

1.  ALL  the  knowledge  with  which  our  senses  furnish  us,  is  either 
of  material  substances  denominated  bodies,  or  of  the  changes  which 
take  place  in  them.     But  there  are  only  two  of  the  senses  that  are 
capable  of  convincing  us  immediately  of  the  existence  of  other  bod- 
ies ;  these  are  the  sight  and  touch  ;  and  the  latter  only  is  capable  of 
determining  with  certainty  whether  an  appearance  is  a  body  or  not. 
Thus  a  body  is  properly  a  palpable  thing. 

2.  There  are  certain  properties  which  are  general,  that  is,  com- 
mon to  all  bodies ;  and  we  are  assured  of  their  generality,  either 
because  we  could  not  perceive  bodies  without  them,  or  because  ex- 
perience has  proved  that  they  are  found  in  all  bodies. 

3.  One  of  the  first  of  these  properties  is  extension,  with  all  its 
modifications.     Every  body  has  a  determinate  form,  although,  in 
liquid  and  aeriform  bodies,  it  varies  with  the  form  of  the  enclosures 
which  confine  them.     Every  body  has  a  determinate  magnitude,  or 
fills  a  certain  space,  which  we  call  its  bulk  or  volume.     Every  body 
is  divisible  ;  and  we  must  distinguish  between  geometrical  and  physi- 
cal divisibility.     The   former  is  unlimited  ;   and  we  are  ignorant 
whether  the  last  be  also  unlimited,  or  whether  there  be  some  point  at 
which  it  ceases ;  experience,  however,  teaches  us  that  by  means  of 
natural  and  artificial  forces,  bodies  may  be  divided  into  particles  so 
small  as  to  become  imperceptible  to  the  senses. 

4.  Another  general  property  is  impenetrability.    By  this  we  mean 
that  no  body  can  coexist  with  another  body,  in  identically  the  sams 
portion  of  space.     It  is  by  impenetrability  that  bodies  become  palpa- 

Ekm.  1 


2  Bodies  in  General. 

ble ;  it  must,  therefore,  belong  to  whatever  we  call  body.  Never- 
theless we  might  doubt,  at  first,  whether  it  exists  in  all  bodies  in  an 
absolute  manner,  that  is,  under  all  circumstances. 

5.  The  property  in  question  certainly  exists;  1.  Between  two 
bodies  perfectly  homogeneous,  of  whatever  kind,  solid,  liquid,  or 
aeriform.     2.  Between   two  solid  bodies,  even  heterogeneous,  so 
long  as  they  continue  in  the  stale  of  solids.     3.  Between  a  solid  and 
a  fluid  body,  so  long  as  the  first  retains  its  solid  state.     Hence  it  is 
evident  that  it  exists  between  our  own  bodies  and  all  other  percepti- 
ble bodies. 

But  this  property  appears  to  be  doubtful  when  two  fluids,  whether 
liquid  or  aeriform,  are  mixed  together ;  or  when  a  solid  body  is  dis- 
solved in  a  fluid  j  or  lastly,  when  two  bodies  form  a  combination  per- 
fectly homogeneous,  which  is  to  be  distinguished  from  a  mixture 
however  perfect.  It  would  seem,  however,  from  a  more  intimate 
knowledge  of  natural  phenomena,  that  impenetrability  exists  as  well 
in  these  cases  as  in  the  other. 

6.  A  necessary  consequence  of  impenetrability  is  coercibility  ;  so 
that  the  words  impenetrable,  palpable,  and  coercible,  signify  the  same 
thing. 

7.  Gravity  and  mobility  are  not  really  necessary  conditions  of 
perceptibility ;  still  experience  teaches  that  they  belong  to  all  per- 
ceptible bodies  without  exception  ;  but  as  these  two  properties  are 
the  most  essential,  considered  with  reference  to  mechanical  phi- 
losophy, we  shall  treat  of  them  at  length  hereafter. 

8.  Most  philosophers  reckon,  also,  among  the  general  properties 
of  bodies,  porosity,  compressibility,  and  elasticity ;  but  the  reasons 
for  the  opinion  that  these  qualities  belong  to  all  bodies,  are  not  con- 
clusive. 

9.  The  preceding  observations  apply  only  to  perceptible  bodies  ; 
but  the  farther  we  advance  in  the  knowledge  of  nature,  the  more  we 
find  ourselves  constrained  to  acknowledge  the  existence  of  imper- 
ceptible substances.     Among  modern  philosophers  there  are  many 
who  incline,  for  good  reasons,  to  deny  the  existence  of  gravity  and 
impenetrability  in  imperceptible  substances,  and  to  allow  them  only 
extension  and  mobility.     Others  ascribe  to  them  the  two  first  pro- 
perues,  but  in  a  degree  imperceptible  to  our  senses.     All  agree  in 
denominating  them  imponderable  or  incoercible  substances* 

*  If  there  really  exist  substances  which  are  not  perceptible    as 
many  phenomena  tend  to  show,  the  ideas  which  we  arc  to  form  of 


State  of  Aggregation  of  Bodies.  3 

CHAPTER  H. 

State  of  Aggregation  of  Bodies. 

10.  ALL  natural  bodies  are  either  solid  or  fluid.     Solid  bodies 
are  those  whose  particles  naturally  adhere  to  each  other,  so  that  they 
cannot  be  separated,  or  change  their  relative  position,  without  the 
action  of  some  external  force ;  which  circumstance  gives  to  these 
bodies  a  particular  and  determinate  form.     In  fluid  bodies,  on  the 
contrary,  the  particles  adhere  so  little  to  each  other,  that  they  may 
be  easily  separated,  and  still  more  easily  change  their  relative  posi- 
tion.    For  this  reason,  these  bodies  cannot  preserve  any  particular 
and  determinate  form.     Among  fluid  bodies,  there  is  a  very  remark- 
able  difference.     Some   naturally  preserve  their  volume,  without 
making  a  continual  effort  to  extend  it ;  these  are  called  liquids.     In 
others,  the  particles  tend  continually  to  separate  from  each  other ; 
these  are  denominated  elastic  or  aeriform  fluids.     The  three  states 
of  aggregation  are  called  solid,  liquid,  and  aeriform  or  gaseous. 

11.  Many  bodies  are  capable  of  passing  successively  into  the  three 
states  of  aggregation,  by  means  of  natural  or  artificial  causes,  without 
experiencing  any  internal  change.     Water,  for  example,  mercury, 
and  most  of  the  easily  fusible  metals,  have  this  property  ;  and  can  be 
successively  rendered  solid,  liquid,  and  aeriform.     Others  appear 
under  two  of  these  states  only ;  of  this  description  are  the  metals 
which  cannot  be  fused  without  great  difficulty  ;  also  certain  liquids 
and  elastic  fluids,  as  alcohol  and  many  other  substances.     Lastly, 
there  are  many  bodies  which  appear  only  in  one  state,  as  the  simple 
earths,  infusible  metals,  and  most  of  the  gases. 

12.  There  are  two  methods  of  changing  the  state  of  aggregation 
of  bodies.     The  first  is  by  heat.     Water,  for  instance,  in  temperate 
climates  is  for  the  most  part  a  liquid.     Below  a  certain  temperature 
it  is  solid  ;  and  above  a  certain  temperature  it  is  aeriform.     The 

their  nature  and  intimate  constitution,  can  only  be  deduced  from  the 
actual  properties,  manifested  by  their  phenomena  ;  which  may  be 
termed  the  abstract  expression  of  the  principle  of  all  their  effects. 
To  this  degree  of  abstraction  we  have  attained  in  the  consideration  of 
electricity  and  magnetism. 


4  Bodies  in  General. 

second  method  is  by  chemical  combination.  When  two  bodies 
combine  very  intimately  together,  one  often  communicates  to  the 
other  its  state  of  aggregation.  Often  also  the  compound  takes  a 
state  of  aggregation  different  from  that  of  the  component  parts.  For 
instance,  salt  becomes  liquid  in  water  ;  water  itself  becomes  an  elas- 
tic fluid  in  air.  Siliceous  earth  passes  to  the  state  of  a  gas  in  fluoric 
acid.  Muriatic  acid  gas  and  ammoniacal  gas  form  the  solid  muriate 
of  ammonia.  Hydrogen  and  oxygen  gases  compose  water  in  a 
liquid  state,  &c.  If,  as  is  very  probable,  heat  be  the  effect  of  an 
imperceptible  substance  capable  of  combining  with  other  substances, 
these  two  methods  are  confounded  together.* 

13.  The  changes  in  the  states  of  aggregation  depend,  according 
to  all  appearances,  upon  the  opposition  of  two  forces,  one  attractive 
and  the  other  repulsive.     The  first  is  a  property  inherent  in  the  par- 
ticles of  bodies.     The  second  is  produced  by  the  caloric  which  com- 
bines with  them.     The  body  is  solid  when  the  attractive  force  ex- 
ceeds the  repulsive  ;  liquid,  when  both  are  in  equilibrium  ;  and  aeri- 
form, when  the  repulsive  force  exceeds  the  attractive. 

14.  The  state  of  aggregation  of  bodies  has  a  great  influence  upon 
all  natural  phenomena  ;  and  particularly  upon  the  laws  of  motion 
and  equilibrium.     It  is  this  which  determines  the  divisions  of  me- 
chanical philosophy  .f 

*  The  hypothesis  here  referred  to  is  far  from  being  well  establish- 
ed. We  know  in  fact,  nothing  at  all  of  the  principle  of  heat.  Many 
phenomena  are  very  well  explained  by  supposing  it  to  be  a  material 
radiating  substance,  capable  of  forming  combinations.  There  are 
others  which  are  less  easily  explained  upon  this  hypothesis  than  upon 
that  of  undulations  excited  in  an  elastic  medium — [See  Cam.  Optics, 
p.  293,  et  seq.] 

t  Aristotle's  opinion  respecting  the  four  supposed  elements,  fire, 
air,  earth,  and  water,  has  an  evident,  though  erroneous  reference  to 
the  different  states  of  aggregation. 


Variety  of  Material  Properties  in  Bodies.  £• 

CHAPTER  ffl. 

Great  Variety  of  Material  Properties  in  Bodies. 

15.  EXPERIENCE  shows  that  bodies  act  differently  upon  one  an- 
other.    In  this  consists  the  material  variety  of  bodies. 

16.  Hitherto  almost  the  only  hypothesis  for  explaining  this  phe- 
nomenon, has  been  the  supposition  that  the  small  particles  of  bodies 
may  have  the  same  material  nature,  and  that  they  differ  only  in 
magnitude,  shape,  and  relative  position,  in  different  bodies ;  but  this 
hypothesis  is  neither  sufficient  nor  probable. 

17.  Experience  is  the  only  sure  guide  in  natural  philosophy  ;  and 
although  the  light  which  it  throws  upon  this  subject,  belongs  to 
chemistry,  it  is  nevertheless  necessary,  in  order  to  convey  a  just  idea 
of  natural  phenomena,  to  give  here  a  brief  view  of  the  results  of 
chemical  analysis. 

18.  Almost  all  bodies  in  nature  are  composed  of  heterogeneous 
substances  ;  thus,  for  example,  cinnabar  is  composed  of  sulphur  and 
oxide  of  mercury  ;   these  substances  are  called  the  constituent  prin- 
ciples, to  distinguish  them  from  the  integrant  particles  which  are 
simply  the  homogeneous  fragments  of  a  body.     Often  the  constitu- 
ent principles  of  a  body  may  themselves  be  decomposed  into  other 
remote  constituent  principles ;  for  example,  the  oxide  of  mercury  is 
composed  of  mercury  and  oxygen.     But  the  chemist  finally  arrives, 
in  all  cases,  at  substances  which  he  cannot  decompose,  either  be- 
cause they  are  really  in  a  simple  undecomposable  state,  or  because 
he  wants  the  means  of  carrying  the  decomposition  farther. 

19.  In  1804,  chemists  reckoned  42  of  these  ponderable  and  un- 
decomposable substances,  of  which  all  bodies  are  formed.    In  1818, 
the  number  exceeded  50.*     Every  elementary  work  on  chemistry 

*  This  number  increases  in  such  a  manner,  as  chemical  researches 
are  multiplied,  that  it  would  be  absolutely  useless  to  attempt  to  fix 
it  with  precision.  We  now  find,  for  example,  in  rough  platina  no 
less  fian  11  simple  or  undecompounded  metals  ;  and  who  can  say 
that  more  will  not  still  be  found  ?  This  multiplicity  is  probably  the 
effect  of  the  little  advancement  yet  made  in  mineral  chemistry  ;  for 
it  cannot  be  presumed  that  these  substances  are  all  strictly  incapable 
of  being  reduced  one  to  another.  But  in  the  present  state  of  the 


6  Bodies  in  General 

contains  a  list  of  these  substances,  which  varies  every  year.     It  now 
comprehends  four  elastic  substances,  and  48  solid  ones, 
last,  four  are'  inflammable,  and  all  the  rest  are  metals.     To  give 
greater  clearness  to  this  subject,  we  shall  consider  some  chemica 
phenomena  more  particularly. 

20.  For  example,  a  pure  salt,  such  as  sulphate  of  soda,  saltpeti 
or  common  salt,  becomes  perfectly  dissolved  in  pure  water,  and 
forms  with  it  a  single  liquid  perfectly  homogeneous.   We  cannot  dis- 
cover, by  the  best  microscope,  a  particle  of  salt  in  this  solution.  This 
phenomenon  seems  to  contradict  the  laws  of  gravity,  if  we  suppose 
that  the  particles  of  salt  only  float  in  the  water,  infinitely  divided. 
We  are  rather  to  believe  that  they  have  themselves  become  fluid, 
and  are  distributed  equally  among  all  the  particles  of  the  water. 
Hence  we  perceive  the  great  difference  there  is  between  a  combina- 
tion and  a  mixture.  The  solution  of  a  salt  is  a  compound  body,  whose 
constituent  principles  are  salt  and  water. 

21.  If  we  expose  such  a  solution  to  a  proper  heat,  the  water 
evaporates ;  but  the  salt  resumes  its  state.     Here  is  an  example  of 
chemical  separation  or  decomposition. 

22.  Sulphate  of  soda  itself  may  be  decomposed  into  sulphuric 
acid  and  soda.     These  substances  are,  therefore,  the  remote  con- 
stituent principles  of  the  saline  solution,  and  the  immediate  constituent 
principles  of  sulphate  of  soda  itself.     But  sulphuric  acid  may  be  de- 
composed into  sulphur  and  oxygen.     Thus,  these  two  last  are,  with 
respect  to  the  sulphate  of  soda,  remote  constituent  principles.     The 
sulphur,  oxigen,  and  soda,  are  simple  substances,  or  as  yet  undecom- 
posed.* 

23.  If  we  mix  one  part  by  bulk  of  oxygen  gas,  called  also  vital 
air,  with  two  parts  of  hydrogen  gas,  or  inflammable  air,  the  result  is 
a  gaseous  mixture  entirely  homogeneous,  called  detonating  gas.     If 

science  it  is  absolutely  necessary  to  regard  as  distinct,  all  substances 
which  resist  our  means  of  decomposition  ;  and  this  is  the  reason  why 
the  number  has  increased  rapidly,  as  chemistry  has  been  extended. 

*  Since  this  work  appeared,  Sir  H.  Davy  has  discovered  that  potash 
and  soda  are  metallic  oxides.  He  arrived  at  this  result  by  causing 
the  electric  current  of  a  powerful  voltaic  battery  to  act  upon  these 
alkalies.  MM.  Guy  Lussac  and  Thenard  have  given  this  fine  dis- 
covery an  important  place  in  chemistry,  by  decomposing  the  alkaline 
oxides  by  the  mere  force  of  affinities. — [On  this  subject,  see  Cami 
Electricity,  p.  165.] 


Variety  of  Material  Properties  in  Bodies.  7 

in  a  thick  glass  vessel,  so  tightly  closed  that  no  ponderable  substance 
can  enter  or  pass  out,  we  place  about  a  fifth  part  of  what  it  is  capa- 
ble of  containing  of  detonating  gas,  and  then  communicate  to  it  the 
electric  spark,  the  enclosed  mass  of  air  becomes  ignited  ;  all  the 
gas  instantly  disappears  and  the  interior  surface  of  the  vessel  is  cov- 
ered with  a  watery  vapour.  By  repeating  the  experiment,  a  suffi- 
cient quantity  of  this  may  be  produced,  to  convince  us  that  the  pro- 
duct is  really  water. 

24.  Water  is,  therefore,  itself  a  compound  body,  and  its  constitu- 
ent principles  are  oxygen  and  hydrogen  ;  but  as  in  the  above  experi- 
ment, nothing  perceptible,  except  the  electric  spark,  was  admitted 
into  the  vessel  or  suffered  to  escape,  and  yet  the  detonating  gas 
underwent  so  remarkable  a  change,  we  are  obliged  to  admit  either 
that  the  electric  spark  separates  from  the  gas,  some  imperceptible 
substance,  which  cannot  be  prevented  from  traversing  the  body  of 
the  vessel,  or  that  it  introduces  some  substance  into  it.*   „ 

25.  In  all  the  chemical  phenomena  above  described,  it  is  certain 
that  nothing  takes  place  but  a  combination  or  a  decomposition  ;  and 
the  same  is  true  of  all  chemical  phenomena.     This  explication  of 
the  material  varieties  of  bodies,  to  which  we  are  led  by  experi- 
ment, consists,  therefore,  in  the   idea  that  there  does  not  exist  a 
great  number  of  simple  substances  ;   but  that  those  which  are  such, 
are  essentially  different,  and  form,  by  their  infinite  combinations,  all 
the  material  differences  which  are  observed  in  bodies. 

26.  The  mixture  of  two  substances  is,  without  doubt,  the  conse- 
quence of  an  attraction,  a  tendency  to  unite,  or  rather  to  penetrate 
mutually  into  the  void  interstices,  which  exist  among  their  material 
particles.     This  is  what  is  meant  by  the  affinity  of  substances  ;  and 
this  property  is  considered  as  a  natural  force  which  is  exerted  at 
each  point  of  contact  of  heterogeneous  bodies,  although  often  it  does 

*  The  author  undoubtedly  refers  to  the  heat  which  is  disengaged 
from  the  mixture,  when  the  two  gases  which  compose  it,  enter  into 
combination  and  form  water.  I  have  proved  by  direct  experiment, 
that  the  transmission  of  the  electric  spark  is  not  necessary  for  the 
formation  of  water.  Enclose  the  two  gases  in  a  gun  barrel  and 
compress  them  rapidly,  and  the  mere  heat  thus  disengaged  will  en- 
flame  them  and  determine  their  combination.  Caution  is  necessary 
in  this  experiment,  for  the  tube  often  bursts  by  the  force  of  the  ex- 
plosion. 


g  Bodies  in  Generdl. 

not  produce  any  combination  between  them,  but  only  a  slight  adhe- 
sion, since  a  more  powerful  force  acts  in  a  contrary  direction  and 
prevents  their  combination. 


CHAPTER  IV. 

Different  Modes  of  Considering  Bodies. 

27.  THE  dynamic  system  prevails  at  the  present  time  in  Germa- 
ny, and  the  atomic  system  in  France.    We  must  not,  therefore,  ne- 
glect to  give  a  brief  exposition  of  these  systems  by  which  human  rea- 
son has  attempted  to  represent  as  far  as  possible,  the  intimate  essence 
of  bodies. 

Atomic  System. 

28.  THE  advocates  of  this  system  suppose  each  body  to  be  com- 
posed of  indivisible  and  impenetrable  particles,  which  they  call  atoms. 
They  are  almost  infinitely  small,  void  spaces  being  interposed  being 
them,  and  thus  porosity  is  rendered  a  necessary  property  of  bodies. 
These  atoms  do  not  touch  each  other,  but  are  kept  at  a  distance  by 
certain  attractive  and  repulsive  forces  which  are  exerted  between 
them.     Hence  it  follows  that  there  is  much  more  void  space  than 
matter  in  any  body.     According  to  this  theory,  the  material  varieties 
of  bodies  may  be  explained  either  by  supposing  a  material  difference 
of  atoms  5  or  a  difference  in  their  form,  magnitude,  position,  and 
distance.     When  two  substances  enter  into  chemical  combination, 
the  atoms  of  one  penetrate  the  interstices  of  the  other,  and  the 
atoms  of  the  two  substances  combine  so  perfectly,  that  they  become 
as  it  were,  a  new  kind  of  constituent  principles,  except  that  they  are 
aot  simple,  but  compound. 


Dynamic  System. 

29.  ACCORDING  to  this  system  every  body  is  considered  as  a  space 
filled  with  continuous  matter.  Porosity  then  becomes  an  accidental 
property  of  matter ;  but  compressibility  and  disability  are  essential 
properties.  The  state  of  a  body  depends  simply  upon  certain  forces 


Empirical  Considerations.  9 

attractive  or  repulsive  ;  and  its  volume  must  change  as  soon  as  the 
ratios  of  these  forces  cease  to  be  the  saine.  According  to  this  theory 
the  material  varieties  of  bodies  are  explained,  by  supposing  the  ex- 
istence of  certain  primitive  substances,  the  different  combinations  of 
which  produce  all  bodies.  When  two  substances  enter  into  chemi- 
cal combination,  the  advocates  of  this  system  are  compelled  to  admit 
that  each  absolutely  penetrates  the  intimate  essence  of  the  other. 


Empirical  Considerations. 

30.  The  history  of  science  teaches  us  that  considerations  pureK 
speculative  have  always  been  productive  of  error.  The  true  philoso- 
pher will  not,  therefore,  yield  implicit  faith  either  to  the  atomic  or 
dynamic  system.  The  intimate  nature  of  bodies  will  always  be  con- 
cealed from  us.  What  we  know  of  them  externally,  we  owe  entirely 
to  an  attentive  observation  and  careful  use  of  what  is  furnished  by 
our  senses.  Mathematics  itself  leads  us  astray,  when  it  has  for  its 
foundation  only  ingenious  hypotheses  and  not  principles  established 
upon  facts.  The  philosopher  ought,  therefore,  to  take  nothing  for 
truth  but  what  has  been  proved  by  experiment.  Still  he  is  allowed 
and  obliged  to  make  use  of  hypotheses  ;  he  must  not  fail,  however, 
to  bring  them  to  the  test  of  observation.  Every  hypothesis  which 
cannot  be  confirmed  or  refuted  by  experiment,  is  nothing  but  a  con- 
ceit or  subtilty.  Nevertheless  hypotheses  of  this  kind  may  some- 
times be  employed  as  means  of  representing  real  things,  but  then  it 
must  always  be  remembered  that  they  are  only  fictions  accommo- 
dated to  the  feebleness  of  our  intellect. 

This  mode  of  studying  nature  is  called  empirical ;  *  and  it  is  re- 
garded as  the  only  exact  means  of  advancing  in  natural  science. 


*  [It  is  more  frequently  called  in  English  the  inductive  method,  and 
Lord  Bacon  is  considered  as  the  first  who  distinctly  recognised  it.] 

EJem.  2 


10  Bodies  in  General 

CHAPTER  V. 

Matliematical  Laws  of  Motion. 

31.  IF  we  conceive  all  bodies  to  be  annihilated,  there  still  remains 
the  idea  of  an  unlimited  extension  in  every  direction  ;  this  is  what 
we  mean  by  infinite  or  absolute  space.     Every  portion  of  this  space 
which  we  conceive  to  be  terminated  in  any  way,  or  which  is  occupi- 
ed by  any  part  of  the  material  universe,  is  called  limited  or  relative 
space.     Absolute  space  is  immoveable,  but  we  can  conceive  of  the 
motion  of  all  relative  space. 

32.  That  portion  of  space  which  is  occupied  by  a  body,  is  called 
its  place.     Motion  is  change  of  place ;  rest  is  continuance  in  the 
same  place.     Both  are  termed  relative  or  absolute,  according  as 
they  are  referred  to  absolute  or  relative  space. 

33.  When  all  the  parts  of  a  body  have  a  common  motion,  the  line 
through  which  any  one  of  its  points  passes  is  called  the  trajectory  of 
the  body.     The  motion  is  rectilinear  or  curvilinear,  according  as  the 
line  described  is  straight  or  curved.     The  motion  is  uniform,  when 
the  body  passes  through  equal  spaces  in  equal  times.     That  which 
is  not  uniform  is  called  varied  motion.     Motion  is  accelerated,  when 
the  spaces  passed  through  in  the  same  intervals  of  time,  become 
larger  and  larger.     It  is  retarded,  when  the  spaces  passed  through  in 
the  same  intervals  of  time,  become  smaller  and  smaller. 

34.  The  space  which  a  body,  moving  uniformly,  passes  through 
in  a  unit  of  time,  a  second,  for  example,  is  called  its  velocity.     In 
the  case  of  motion  which  is  not  uniform,  the  velocity  changes  each 
instant ;  and  this  velocity,  in  a  given  instant,  is  equal  to  the  space 
which  the  body  would  describe  in  a  unit  of  time,  if  it  were  uniformly 
to  preserve  the  velocity  which  it  then  had. 

35.  In  the  case  of  uniform  motion  the  space  described  is  propor- 
tional to  the  time.     We  ascertain  the  velocity,  therefore,  by  dividing 
the  space  by  the  time  employed  in  describing  it.* 

•*  Let  S  denote  the  space  described  by  a  body  moving  uniformly  in 
the  time  T.    If  we  designate  the  velocity  by    Vt  we  shall  have 

V=  T'    T*"s  fundamental  formula  of  all  mechanics  serves  also  for 


Mathematical  Laws  of  Motion.  11 

36.  With  respect  to  absolute  space,  a  body  can  have  only  one 
motion  at  the  same  time  ;  but  with  respect  to  relative  space,  it  may 
have  an  indefinite  number ;  for  if  a  body  has  a  motion  with  respect 
to  a  relative  space,  this  space  may  have  a  second  motion  with  re- 
spect to  another  relative  space,  which  may  itself  have  a  third  motion 
with  respect  to  the  body  and  the  first  space  ;  and  so  on. 

37.  If  a  body  situated  in  A,  (fig.  1)  have  two  uniform  motions  in 
the  directions  AB  and  AC,  and  if  AB  and  AC  represent  the  spaces 
which  would  be  described  in  equal  times,  by  virtue  of  these  separate 
motions,  the  body,  by  virtue  of  the  two  combined,  will  describe  in  the 
same  time,  and  ivith  a  uniform  motion,  the  diagonal  AD  of  the  par- 
allelogram ABCD,  which  is  constructed  upon  the  lines  AB  and  AC. 

According  to  the  preceding  article  there  is  only  one  way  of  caus- 
ing the  body,  placed  at  A ,  to  take  two  motions  at  the  same  time ;  and 
that  is  by  giving  it  a  motion  in  relative  space,  and  then  giving  another 
motion  to  this  relative  space  itself,  together  with  the  body.  Let  A  C, 
therefore,  be  the  line  which  the  body,  moving  uniformly,  describes  in  a 
given  time,  in  the  relative  space  in  which  it  moves ;  and  at  the  same 
time,  let  A  C  itself  be  moved  uniformly  in  the  direction  AB,  in  such  a 
manner  that  at  the  expiration  of  the  given  time,  it  shall  occupy  the 
situation  BD.  It  is  easy  to  conceive  that  the  body,  by  the  combi- 
nation of  these  two  uninterrupted  motions,  will  describe  the  diagonal 
AD,  and  that  it  will  describe  it  with  a  uniform  motion.* 

If  we  cause  the  body  to  move  in  the  direction  AB,  and  the  line 

infinitely  small  motions,  provided  S  and  T  be  regarded  as  infinitely 
small. 

In  this  equation  S  does  not  represent  a  line,  but  the  number  of 
linear  units,  feet  for  example,  which  have  been  described ;  and  in 
the  same  manner  T  represents  the  number  of  units  of  time,  as  sec- 
onds for  example,  employed  in  describing  them.  Accordingly  S  and 
T  and  their  quotient  V,  are  abstract  numbers.  In  general,  we  cannot 
bring  heterogeneous  quantities,  as  space  and  time,  into  direct  com- 
parison. Each  must  first  be  reduced  to  units  of  its  own  kind,  and 
then  we  have  only  to  compare  abstract  numbers.  The  same  is  true 
in  all  cases  when  physical  data  are  introduced  into  calculation. 

*  These  two  propositions  are  capable  of  being  rigorously  demon- 
strated, according  to  the  mathematical  principles  of  mechanics;  that 
is,  we  deduce  them  from  the  abstract  idea  of  forces — [See  Cam. 
Mf.ch.  art.  35,  et  seq.] 


12  Bodies  in  General. 

AB  to  pass  to  the  situation  CD,  the  consequence  will  be  the  same. 

38.  We  call  the  motions  AB  and  AC  simple  or  lateral  motions, 
and  AD  the  mean  or  compound  motion.     This  very  important  pro- 
position is  called  the  theorem  of  the  composition  of  motion. 

39.  It  makes  no  difference  whether  we  say  the  body  has  the  two 
motions  AB  and  AC.  or  only  the  single  motion  AD.  We  may  com- 
pound the  two  motions  in  one,  and  reciprocally  we  may  decompose 
each  motion  AD,  into  two  others  AB  and  AC,  in  directions  taken 
at  pleasure. 

40.  By  repeating  the  application  of  the  theorem,  any  number 
whatever  of  motions  may  be  compounded  into  one ;  or  a  single  mo- 
tion may  be  decomposed  into  any  number  of  motions  in  directions 
taken  at  pleasure. 

41.  This  theorem  may  even  be  applied  to  motions  in  curved  lines 
and  to  those  which  are  not  uniform,  if  we  represent  by  AB  and  AC, 
infinitely  small  spaces,  which  may  be  described  in  infinitely  small 
times.     This  proposition  may,  therefore,  be  extended  to  all  sorts  of 
motions  which  can  be  imagined. 

42.  Every  absolute  motion  may  be  considered  as  relative,  if  we 
refer  it  to  a  limited  space.     Every  relative  motion  may  be  consider- 
ed as  absolute,  if  we  regard  the  relative  space  as  at  rest.     It  is, 
therefore,  of  no  importance  to  us,  for   any  physical  purpose,  to 
know  whether  a  body  in  motion  or  at  rest,  is  so  absolutely  or  rela- 
tively. 


CHAPTER  VI. 

Physical  Laws  of  Motion,  or  Investigation  of  Moving  Forces. 

43.  EVERY  motion,  and  every  change  produced  in  the  velocity  or 
direction  of  a  motion,  must  have  a  cause,  like  any  other  change  of 
state  m  bodies.  That  principle,  whatever  it  is,  which  we  recognise  as 
the  immediate  cause  of  a  change  in  the  state  of  rest  or  motion  of  a 
body,  is  called  a  moving  force. 

comPrised  "^er  'he  follow- 


Physical  Laws  of  Motion.  13 

(1.)  The  volition  of  animated  beings  is  capable  of  producing 
motion  by  means  of  the  muscles  of  the  body.  This  is  the  only  kind 
of  moving  force  which  we  recognise  immediately  by  sensation. 

(2.)  The  mobility  of  all  bodies,  taken  in  connexion  with  their  im- 
penetrability, results  in  a  moving  force  ;  for  if  two  impenetrable 
bodies  impinge  against  one  another,  a  change  must  necessarily  take 
place  in  their  respective  states ;  that  is,  they  exert  opposite  forces 
against  each  other. 

(3.)  Moving  forces  also  exist  in  the  particular  properties  of  many 
bodies  ;  among  others,  in  the  elasticity  of  solid  bodies,  and  the  dilata- 
bility  of  aeriform  bodies. 

(4.)  There  are  many  motions  of  which  we  do  not  know  the 
cause,  or  only  know  it  very  imperfectly.  This  is  the  case  with  the 
motions  produced  by  gravity,  magnetism,  heat,  electricity,  &c.* 

45.  In  reality,  it  is  impossible  to  withdraw  a  body  physically  from 
the  influence  of  all  moving  forces.     But  we  can  imagine  it  thus  with- 
drawn ;  and  this  is  necessary  in  order  to  apply  the  theory.     Then 
there  will  remain  only  the  idea  of  an  inert  mass,  deprived  of  all 
force,  and  incapable  of  changing  its  state.     This  being  supposed,  the 
first  law  to  which  Newton  reduced  the  theory  of  motion  was  the  fol- 
lowing :  A  body  at  rest  continues  in  a  state  of  rest,  and  a  body  put 
in  motion,  continues  in  a  state  of  motion,  uniformly  and  in  a  straight 
line,  until  some  moving  force  changes  the  state  supposed.     This  in- 
difference to  motion  and  rest,  is  regarded  as  a  general  property  of 
bodies,  and  is  called  the  force  of  inertia. 

46.  The  mass  of  a  body  is  the  quantity  of  matter  which  it  contains. 
This  must  not  be  confounded  with  its  bulk  or  volume.     In  the  chap- 
ter on  gravity,  we  shall  see  that  the  weight  of  a  body  is  the  measure 
of  its  mass. 

47.  The  quantity  of  motion  depends  in  part  upon  the  mass  of  the 
body  put  in  motion,  and  in  part  upon  its  velocity.     If  die  masses  are 
equal,  the  quantities  of  motion  are  proportional  to  the  velocities ;  if  the 
velocities  are  equal,  to  the  masses.     Hence  it  follows  that  generally 
the  quantity  of  motion  is  as  the  product  of  the  mass  into  the  velocity.-}- 


*  The  motions  imparted  to  material  bodies  by  the  electric  and 
magnetic  forces,  are  now  perfectly  explained  in  all  their  details. 

t  Let  the  masses  of  the  two  bodies  in  motion  be  M  and  m ;  their 
respective  velocities  V  and  v  ;  and  their  respective  quantities  of 
motion  B  and  b.  If  then  we  suppose  another  body  which  has  the 


14  Bodies  in  General. 

48.  Since  we  have  little  or  no  knowledge  of  moving  forces,  con- 
sidered in  themselves,  we  must  be  equally  ignorant  of  the  imme- 
diate measure  of  these  forces ;  but  we  can  measure  them  by  their 
effects ;   that  is,  by  the  amount  of  the  motions  which  they   pro- 
duce ;  and  we  know  that  the  force  employed  must  be  proportional 
to  it.     A  force,  is  therefore  measured  by  the  product  of  the  mass 
into  the  velocity  of  the  body  in  motion  (47.)     Such,  in  substance,  is 
Newton's  fundamental  law  of  motion.* 

49.  It  is  obvious  from  this,"  in  what  manner  moving  forces  may  be 
represented  by  numbers  or  lines.     The  latter  method  is  particularly 
convenient,  when  two  or  more  forces  act  upon  the  same  body  ;  then 
the  lines  which  express  the  ratios  of  the  forces,  represent  at  the  same 
time  their  directions,  the  ratios  of  the  velocities  which  they  tend  to 
communicate  to  the  body,  and  lastly,  the  ratio  of  the  spaces  which 
the  body  would  have  described,  if  it  had  been  moved  by  each  of  the 
forces  separately. 

50.  The  third  fundamental  law  of  motion,  discovered  by  Newton, 
is  the  following.      When  two  bodies  act  one  upon  the  other,  t/ieir  ac~ 
tions  and  reactions  are  always  equal ;  that  is,  if  the  motion  of  a  body 
becomes  itself  a  moving  force,  by  the  pressure  or  impulse  which  it 
produces  upon  the  other  body,  the  two  bodies  experience  an  equal 
but  opposite  effect.     What  one  gains  in  motion  the  other  loses,  since 
the  force  must  be  equal  to  its  effect  (46.) 

same  mass  M  as  the  one,  and  the  same  velocity  v  as  the  other ;  and 
call  its  quantity  of  motion  £ ;   we  shall  have  the  proportion 

B:/*::F:»; 
whence,  multiplying  the  two  proportions  together,  we  have 

B:b::MV :  m  v. 

*  This  proportionality  is  in  itself  neither  evident  nor  necessary. 
We  can  conceive  of  an  infinite  number  of  mechanical  laws  of  mo- 
tion ;  in  which  the  velocity  will  not  be  proportional  to  the  moving 
force.  But  the  phenomena  which  must  result  from  them  in  the  com- 
position of  motion,  would  differ  from  those  which  the  present  state 
of  the  universe  presents  to  us  ;  and  the  latter  tak«  place  as  if  the 
velocity  were  proportional  to  the  force.  This  law,  therefore,  is  the 
only  one  which  ought  to  be  physically  admitted ;  though  it  is  obvi- 
ous that  its  truth  is  contingent.— See  La  Place's  Systeme  du  Monde 
and  Mecanique  Celeste. 


Gravity.  15 

CHAPTER  VII. 

Historical  View  of  what  is  known  respecting  Gravity. 

51.  GRAVITY,  considered  with  respect  to  its  effects,  is  by  far  the 
most  important  of  natural  mechanical  forces.  Its  cause  is  entirely  un- 
known.    But  we  know  the  laws  by  which  it  operates  more  exactly 
than  those  of  any  other  natural  force.  As  the  illustration  of  its  effects 
is  one  of  the  principal  objects  of  mechanical  philosophy,  it  is  proper 
that  they  should  be  described  in  all  their  details.     Most  of  these 
effects,  however,  can  only  be  stated  here  in  a  historical  manner  ;  for 
some  of  the  methods  by  which  philosophers  have  determined  the  laws 
of  gravity,  must  necessarily  be  deferred  till  the  reader  is  further  ad- 
vanced ;  and  the  rest  do  not  belong  to  the  present  subject,  but  to 
astronomy. 

52.  The  first  effect  of  gravity  which  we  have  to  consider  is  the 
pressure  directed  towards  the  earth,  which  each  body  exerts  upon 
those  which  are  placed  beneath  it.     This  pressure,  the  determinate 
intensity  of  which  is  called  the  weight  of  a  body,  may  be  very  exactly 
measured  by  means  of  a  balance.    It  is  invariable,  whatever  changes 
take  place  in  the  form,  position,  extension,  and  chemical  properties 
of  a  body,  provided  that  no  ponderable  matter  is  added  or  taken  away. 
This  circumstance  justifies  the  conclusion  that  the  weight  of  a  body 
depends  solely  upon  the  quantity  of  ponderable  matter  which  it  con- 
tains, and  consequently  that  the  mass  must  be  in  the  same  proportion.* 

53.  Experience  teaches  us  that  when  several  bodies  are  homoge- 
neous, that  is,  absolutely  identical  as  to  their  nature  and  constitution, 
the  weights  of  these  bodies  are  to  each  other  as  their  bulks.     But 
this  proportionality  does  not  hold  with  respect  to  bodies  which  are 
heterogeneous,  either  by  nature,  or  in  consequence  of  the  circum- 
stances in  which  they  are  placed.     Hence  arises  what  is  termed  the 

*  In  order  that  nothing  may  enter  into  the  definition  of  the  mass, 
which  depends  upon  the  constitution  of  bodies,  we  arc  to  understand 
by  material  points  equal  in  mass,  those,  which,  having  equal  and  op- 
posite velocities,  would  be  an  equilibrium.  The  same  definition  ex- 
tends also  to  equal  masses.  For  example,  this  equality  takes  place 
with  respect  to  the  masses  which  are  in  equilibrium  in  the  two  scales 
of  a  balance,  the  arms  of  which  are  equal. 


1§  Bodies  in  General, 

density,  or,  iu  other  words,  the  specific  or  proper  gravity  of  a  body. 
Specific  gravity,  then,  is  the  ratio  of  the  absolute  weight  of  one  body 
to  that  of  another,  the  latter  being  taken  for  unity. 

54.  In  estimating  the  specific  gravity  in  numbers,  we  employ  two 
kinds  of  units.     For  solids  and  liquids,  the  weight  of  pure  water  is 
taken  for  unity ;  we  weigh  a  body  of  a  given  bulk,  and  determine 
the  weight  of  the  same  bulk  of  water.    We  divide  the  first  weight  by 
the  second,  and  thus  obtain  the  specific  gravity  of  the  body.*     This 
estimate  has  the  advantage  of  being  independent  of  all  differences  of 
weights  and  measures.  There  are  many  means  of  estimating  the  spe- 
cific gravities  of  bodies.     The  most  exact  and  best  methods  will  be 
explained,  when  we  come  to  treat  of  hydrostatics. 

55.  In  the  case  of  aeriform  bodies,  we  commonly  take  the  weight 
of  a  cubic  inch  of  the  gas  itself  and  compare  it  with  the  same  bulk 
of  atmospheric  air.     But  this  mode  of  estimating  is  subject  to  varia- 
tion on  account  of  a  difference  in  the  weights  and  measures  em- 
ployed.    We  shall  speak  of  the  methods  made  use  of  under  the  head 
of  areometry. 

56.  When  a  heavy  body  is  not  sustained,  it  falls  with  an  accele- 
rated motion,  the  laws  of  which  we  shall  soon  undertake  to  investi- 
gate.    The  direction  of  the  descent  is  called  a  vertical.     A  line  or 
surface  to  which  this  direction  is  perpendicular  is  called  a  horizontal 
line  or  surface.     This  is  the  direction  always  assumed  by  the  sur- 
face of  liquids  in  a  tranquil  stale. 

57.  When  we  compare  the  directions  of  gravity  in  places  very 
near  each  other,  they  appear  to  be  absolutely  parallel ;  but  a  more 
exact  knowledge  of  the  terrestrial  globe  has  taught  us  that  they  are 
everywhere  so  directed  as  to  meet  at  points  near  the  centre  of  tin- 
earth. 

58.  In  a  vacuum,  or  space  void  of  air,  all  bodies  fall  with  the  same 
velocity.     This  law  is  well  confirmed  by  the  theory  of  the  pendulum, 
and  furnishes  a  rigorous  demonstration  that  the  masses  of  bodies  are 
proportional  to  their  weights.     A  body  falling  without  obstruction 

*  Let  JFbe  the  absolute  weight  of  a  body,  expressed  in  units  of  a 
determinate  kind,  as  ounces ;  let   A  be  the  absolute  weight  of  an 
equal  bulk  of  water,  expressed  in  the  same  manner.     If  we  call  S  thf 
specific  gravity  of  the  body,  we  shall  have  the  proportion 
A:W::i:S',  whence  S  =  ~. 


Gravity.  17 

describes  in  the  first  second  a  space  equal  to  16,1  feet.     This  also 
is  ascertained  by  the  theory  of  the  pendulum. 

59.  In  the  same  place  gravity  is  invariable.     Observations  on  the 
pendulum  have  confirmed  the  position  of  Newton,  that  gravity  cannot 
be  the  same  at  every  point  of  the  earth's  surface,  and  that  its  inten- 
sity is  less  at  the  equator  than  at  the  poles.     This  variation  necessa- 
rily results  from  the  circumstance,   that  the  earth  is  not  exactly 
spherical.     But  as  the  quantity  by  which  its  figure  varies  from  a 
sphere  is  very  inconsiderable,  the  inequality  thus  produced  in  gravity, 
is  likewise  very  small. 

60.  Gravity  has  been  found  to  be  a  little  less  on  very  high  moun- 
tains than  on  plains.     This  observation  would  naturally  have  led  us 
to  the  inference  that  gravity  decreases  as  we  remove  from  the  surface 
of  the  earth,  if  Newton  had  not  already  made  the  discovery  in  an- 
other way. 

61.  Newton,  by  a  profound  knowledge  of  the  general  laws  of 
motion,  and  availing  himself  of  what  the  researches  and  observations 
of  two  thousand  years  had  taught  respecting  the  motions  of  the 
heavenly  bodies,  demonstrated  that  a  reciprocal  attraction  exists  be- 
tween all  bodies  in  nature,  and  that  it  is  in  the  direct  ratio  of  the 
mass  of  the  attracting  body,  and  in  the  inverse  ratio  of  the  square  of 
of  its  distance.     Then,  comparing,  by  means  of  the  calculus,  the 
force  which  retains  the  moon  in  its  orbit,  with  the  force  which  causes 
bodies  here  to  fall  towards  the  surface  of  the  earth,  he  proved  that 
this  last  gravity  is  nothing  else  than  a  particular  case  of  that  attrac- 
tion, common  to  all  bodies,  which  he  called,  for  this  reason,  universal 
gravitation. 

62.  Newton  has  shown,  that  the  force  of  gravity  takes  place  in 
some  of  the  most  important  phenomena  of  the  universe.     It  is  this 
which  binds  together  all  the  several  parts  of  a  body,  in  such  a  man- 
ner, that  no  ponderable  particle  can  ever  be  lost.     It  is  this  which 
unites  in  one  vast  whole  all  the  bodies  of  which  the  planetary  sys- 
tem is  composed,  imparting  a  perpetual  order  and  harmony  to  their 
motions.    If  the  Creator  were  to  break  this  invisible  chain,  all  nature 
would  return  to  a  state  of  chaos. 


SECTION  II. 

SOLID   BODIES. 


CHAPTER  Vm. 

General  Properties  of  Solid  Bodies. 

63.  THE  parts  of  a  solid  body  adhere  together  in  such  a  manner 
that  an  effort  is  required  to  separate  them,  or  even  to  change  their 
respective  positions.  On  this  account  a  solid  body  has  a  particular 
determinate  form.  The  force  which  connects  its  different  parts,  is 
called  the  force  of  cohesion.  Musschenbroek,  Buffon,  and  some 
others,  have  jnade  numerous  experiments  upon  this  force. 

Musschenbroek  caused  bars  of  metal  to  be  formed  0,17  of  an 
inch  square,  and  suspended  by  them  different  weights  in  succession 
until  he  broke  them.  The  following  is  the  result  of  his  experiments 
upon  different  metals  ; 

lb. 

For  German  Iron  .  •'"".'       .         .         .  1930 

Fine  Silver  .         .         .      '  .         .         .  H56 

Swedish  Copper  .         .         .         .      '  .  1054 

Fine  Gold  .         .                  .         .         .  573 

English  Tin  .         .         .  ~    ';V"  ^-  •?•••'  150 

Malacca  Tin  .         .         .         ....  91 

English  Lead  .         .         ,....,  25 

Similar  experiments,  made  with  rods  of  wood  0,28  of  an  inch  square, 
gave  the  following  results  ; 

For  Beech  .         .         .  '       i 


Oak  •         •         •.-.".         .         .  1150 

Linden         .         .         ...         .  ]000 

£"•  •'.....  600 

Pme    '         •         •         >.".'.'  550 


General  Properties  of  Solid  Bodies.  19 

The  force  of  cohesion  in  ductile  metafs  is  increased  by  moderate 
hammering ;  but  too  heavy  blows  diminish  it.  It  is  diminished  by 
heat  and  increased  by  cold.  In  general,  cohesion  is  a  variable  force, 
capable  of  being  altered  by  a  great  many  chemical  and  mechanical 
means. 

64.  The  property  which  particles  have  of  changing  their  relative 
positions  without  being  disunited,  is  not  always  proportional  to  their 
susceptibility  of  being  separated.     This  distinction  gives  rise  to  the 
various  qualities  of  solid  bodies,  which  we  designate  by  the  some- 
what vague  terms,  hard,  soft,  tenacious,  friable,  stiff,  flexible,  &tc. 

65.  The  force  of  cohesion  still  manifests  itself  when  bodies  are 
divided  into  small  fragments.     It  appears  that  all  bodies  when  placed 
in  contact,  have  a  tendency  to  attach  themselves  to  one  another ; 
unless  the  contact  is  very  imperfect,  or  the  weight  of  the  bodies  ren- 
ders the  effect  insensible.     This  property  is  called  adhesion.     It  is 
distinguished  from  cohesion,  because  it  exerts  itself  as  well  upon 
heterogeneous   as   upon  homogeneous  bodies  ;    whereas  cohesion, 
properly  speaking,  only  takes  place  between  homogeneous  bodies.* 

66.  The  attraction  which  exists  between  two  homogeneous  bodies, 
is  properly  called  affinity.     Adhesion  is  not,  therefore,  a  distinct 
force,  but  may  be  considered  as  a  feeble  cohesion  or  a  feeble  affinity. 

67.  We  now  know  three  sorts  of  attractions  which  act  upon  all 
bodies.     These  are,  gravity,  cohesion,  and  affinity.     The  cause  of 
these  attractions  is  entirely  unknown  ;  we  are  even  ignorant  whether 
they  depend  upon  one  or  several  causes.    But  the  phenomena  which 
they  produce  are  so  different,  that  we  are  obliged  to  attribute  them 
to  different  causes  until  it  is  ascertained  that  they  act  according  to 
one  single  law.** 

*  Adhesion  and  cohesion  ought  not  to  be  regarded  as  distinct 
forces,  but  rather  as  modifications,  or  simply  as  analogous  effects  of 
that  affinity  which  attracts  .particles  towards  each  other. 

**  It  is  very  probable  that  the  laws  of  chemical  affinity  differ  from 
the  law  of  gravity.  This  last  is  reciprocally-  proportional  to  the 
square  of  the  distance  ;  and  without  knowing  the  nature  of  affinity, 
it  will  be  seen  that  its  action  decreases  much  more  rapidly,  so  that  it 
ceases  to  be  sensible  at  very  small  distances.  It  is  true  that  a  force 
reciprocally  proportional  to  the  square  of  the  distance,  when  it  acts 
upon  the  different  points  of  the  body,  may  produce  results  depend- 
ing on  its  figure,  and  decreasing  very  rapidly  in  intensity.  It  is  thus 


20  Solid  Bodies. 

68.  Compressibility  seems  to  be  a  property  common  to  all  solid 
bodies.     But  the  effect  of  compression  is  so  feeble  in  very  solid 
bodies,  that  we  may  most  generally  regard  it  as  nothing.     Still  even 
in  this  case  we  can  conceive  of  its  existence. 

69.  Some  bodies  preserve,  after  the  pressure  ceases,  the  form 
which  they  have  thus  received  ;  others  tend  to  recover  their  primi- 
tive figure  and  to  fill  their  primitive  space,  the  moment  the  pressure 
is  removed.     These  last  are  called  elastic.     Elasticity  has  a  great 
influence  upon  the  phenomena  of  motion,  and  we  ought,  for  this 
reason,  to  give  a  more  precise  idea  of  it. 

70.  An  elastic  body  always  tends  to  recover  its  form  and  bulk. 
If  it  is  pressed  in  one  part,  it  distends  itself  in  another,  where  it  can 
act  freely.     If  it  is  enlarged  in  one  part,  it  contracts  in  another.     If 
it  is  bent  and  then  left  to  itself,  it  returns  at  once  to  its  former  dis- 
position. 

71.  There  appear  to  be  no  bodies  perfectly  elastic  or  perfectly 
unelastic.     Every  solid  possesses  this  property  to  a  greater  or  less 
degree.     Among  the  most  elastic  bodies,  we  may  reckon  tempered 
steel,  bell-metal,  ivory,  bone,  dry  wood,  gum  elastic,  &c.     Among 
the  least  elastic,  on  the  contrary,  may  be  mentioned  the  softer  metals, 
tin,  lead,  gold,  silver,  soft  clay,  &tc.     This  property  of  solid  bodies 
must  not  be  confounded  with  the  dilatability  of  aeriform  bodies, 
which  is  also  sometimes  called  elasticity.* 

72.  It  is  obvious  from  mere  inspection  that  many  solid  bodies  are 
porous  ;  others  are  ascertained  to  be  so  from  experiment ;  but  there 
are  also  many  solid  bodies,  such  as  glass,  well  cast  metals,  &c.,  the 
porosity  of  which  cannot  be  demonstrated  either  by  the  magnifying- 
glass,  or  any  other  physical  means.    Moreover,  in  many  bodies  there 
are  phenomena  which  seem  absolutely  to  contradict  the  idea  of  po- 

that  the  precession  of,  the  equinoxes,  produced  by  the  attraction  of 
the  sun  and  moon  upon  the  oblate  spheroid  of  the  earth,  decreases, 
as  the  cube  of  the  distance  increases ;  that  is,  it  would  be  eight 
times  smaller  if  these  bodies  were  twice  as  far  apart.  But  it  is  diffi- 
cult to  suppose  that  there  exists  no  other  essential  difference  than 
that  of  figure,  between  particles  of  such  different  properties,  as  those 
which  compose  bodies. 

*  The  phenomena  of  elasticity  have  been  the  subject  of  very  valu- 
able experiments  made  by  s'Gravesande,  and  afterwards  by  Coulomb. 


Interior  Construction  of  Solid  Bodies.  21 

rosity.*    Accordingly,  the  dynamic  theory  does  not  suppose  porosity 
necessary  to  the  explanation  of  the  different  densities  of  bodies. 


CHAPTER  IX. 

The  Interior  Construction  of  Solid  Bodies. 

73.  MOST,  if  not  all  solid  bodies,  appear  to  form  a  certain  regular 
assemblage  of  small  parts.    The  regularity  of  the  fragments  of  many 
minerals,  the  superposition  of  the  laminae  in  transparent  crystals,  the 
facility  with  which  many  bodies  are  broken  or  cleft  in  certain  direc- 
tions rather  than  in  others,  the  breaking  of  unannealed  glass  drops, 
present  phenomena  which  confirm  this  remark. 

74.  The  discoveries  of  M.  Haiiy  give  us  the  most  exact  idea  of 
the  structure  of  crystals.     This  idea  is  the  more  important  because 
crystallization  appears  to  be  a  general  law  of  nature.    Crystallization 
consists  in  this,  that  in  the  passage  from  the  fluid  to  the  solid  state, 
each  body  takes  a  regular  and  determinate  form. 

75.  According  to  the  researches  of  M.  Haiiy,  every  crystal,  form- 
ed naturally  or  artificially,  admits  of  being  cleft  and  subdivided, 
according  to  the  direction  of  certain  planes,  with  more  facility  than  in 
any  other  way.     This  operation  is  different  for  different  crystals.    If 
it  be  continued  in  all  the  directions  where  it  is  possible,  the  laminae 
which  result  from  it  being  successively  removed,  until  none  of  the 
exterior  surface  remains,  we  obtain  a  nucleus,  which  ordinarily  has  a 
different  form  from  that  of  tne  entire  crystal,  and  subsequent  divis- 
ions of  it  would  only  diminish,  without  changing  the  direction  of  the 
planes  which  terminate  it.     M.  Haiiy  calls^  the  form  of  the  entire 
crystal,  the  secondary  form,  that  of  the  nucleus  the  primitive  form. 
The  nucleus  itself  may  be  divided  iuto  corpuscles  of  the  same  form, 
which  he  calls  integrant  particles.     The  detached  laminae  are  also 
composed  of  similar  particles  as  their  cleavage  proves,  and  conse- 
quently the  entire  crystal  is  formed  of  them.     M.  Haiiy  supposes 

*  It  is  probable  the  author  speaks  here  only  of  what  appears,  and 
not  of  what  is  in  reality  the  case  ;  for  I  cannot  believe  that  there  are 
any  phenomena  which  are  not,  in  reality,  strictly  reconcilable  with 
the  idea  of  porosity. 


22  Solid  Bodies. 

that  these  integrant  particles  are  themselves  small  crystals,  the  con- 
figuration of  which  is  determined  by  the  combination  of  the  elemen- 
tary particles  which  constitute  the  rudiments  of  the  entire  crystal. 
As  yet  there  has  been  no  experiment  to  verify  this  suPP°sltlon- 

76  Hitherto,  M.  Haiiy  has  found  only  three  form  of  integrant 
particles  ;  the  triangular  pyramid,  the  triedral  prism,  and  the  quad- 
rangular prism.  He  has  ascertained  only  six  primitive  forms  ;  the 
parallelopiped,  octaedron,  tetraedron,  hexaedral  prism,  rhomboidal 
dodecaedron,  and  triangular  dodecaedron.  The  secondary  forms 
vary  without  end,  by  the  regular  aggregation  of  the  primitive  forms. 

77.  It  is  extremely  remarkable  that  crystals  of  the  same  substance 
have  a  nucleus  and  integrant  particles  of  the  same  forms,  while  on 
the  contrary,  their  exterior  figure  admits  of  an  endless  variety,  as  is 
often  realized  to  a  great  degree  in  the  productions  of  nature.* 

78.  The  regular  form  of  crystals  and  the  property  of  being  easily 
divided  in  certain  directions,  indicate  that  the  force  of  cohesion  does 
not  exert  its  power  to  the  same  degree  in  all  the  points  of  their 
particles ;  and  that  these  particles  have  certain  poles  of  attraction, 
which,  according  to  their  greater  attractive  force,  determine  the  posi- 
tion of  these  particles.** 

79.  When,  by  artificial  means,  we  cause  fluid  bodies  to  pass  to  the 
state  of  solids,  they  always  form  themselves  into  regular  crystals ; 
only  these  crystals  are  sometimes  so  small,  that  a  microscrope  is 
necessary  to  enable  us  to  perceive  them.     If,  as  is  by  no  means  im- 
probable, all  solid  bodies  have  first  passed  through  the  fluid  state,  we 
are  induced  to  believe,  that  an  exact  and  most  minute  division  would 
make  known  a  similar  regularity  in  their  interior  structure. 

For  a  better  understanding  of  the  theory  of  crystallization,  the  rea- 
der is  referred  to  the  Mineralogy  of  M.  Haiiy. 


*  Arragonite  appears  to  form  an  exception  to  this  rule.  Its  crys- 
tallization differs  from  that  of  calcareous  spar,  though  its  chemical 
principles  appear  to  be  the  same. 

**  There  is  a  certain  number  of  crystals  whose  cleavage  is  equally 
easy  or  difficult  in  all  directions  where  it  is  possible  ;  if,  moreover, 
the  faces  of  cleavage  are  equally  smooth  and  of  a  similar  aspect,  we 
regard  this  equality  as  the  indication  of  a  symmetry  in  the  corre- 
sponding faces  of  the  primitive  particle.  In  general,  the  direction  of 
the  planes  of  cleavage  and  the  characters  of  the  faces,  compose  what 
is  called  the  crystalline  system,  and  constitute  indications  to  which 
the  integrant  particle  is  conformed. 


Equilibrium  of  free  Bodies.       .  23 

CHAPTER  X. 

Equilibrium  of  Solid  Bodies,  or  Fundamental  Principles  of  Statics. 

80.  WE  have  here  two  cases  to  examine  ;  1.  That  in  which  the 
forces  act  upon  a  free  body,  that  is,  one  which  is  not  obstructed  in 
any  way.     2.  That  in  which  they  act  upon  a  body  which  is  not  free, 
but  so  confined  as  to  move  about  a  certain  point  or  about  an  axis. 

Equilibrium  of  Free  Bodies. 

81.  When  a  material  point  is  acted  upon  at  the  same  time  by  two 
equal  but  directly  opposite  forces,  as  AB,  AC,  (Jig.  2.)  this  point 
cannot  obey  either.     In  this  case,  the  state  of  rest  which  results  from 
the  mutual  destruction  of  the  two  forces,  is  called  equilibrium.  If  the 
opposite  forces  are  unequal,  there  results  a  motion  in  the  direction  of 
the  greater. 

82.  When  two  forces,  AB  and  AC,  (jig.  3.)  act  in  different  direc- 
tions upon  the  body  A,  it  cannot  obey  both  at  the  same  time,  except 
by  describing  the  diagonal  AD,  of  the  parallelogram  ABCD.     The 
two  forces,  therefore,  have  exactly  the  same  effect  as  the  single  force 
AD  would  have. 

83.  Upon  this  is  founded  the  composition  and  decomposition  of 
moving  forces.     Instead  of  the  two  forces  AB  and  AC,  we  may 
find  the  single  force  AD,  which  is  equivalent  to  them.     AB  and 
AC  are  called  the  simple  forces  or  components  ;  and  AD  the  result- 
ant.    Reciprocally,  we  can  decompose  each  simple  force  AD,  into 
two  forces  AB  and  AC,  taken  in  any  given  directions,  and   acting 
precisely  as  this  force.     Hence  the  phrase  parallelogram  of  forces. 
It  is  easy  to  perceive  how  two  or  more  forces  may  be  resolved  into 
one,  and  how  one  may  be  decomposed  into  two  or  more. 

84.  If  three  forces,  AB,  AC,  AE,  (Jig.  3.)  keep  the  body  A  in 
equilibrium,  the  third,  AE,  must  be  in  the  direction  of  the  diagonal 
AD  produced ;  and  it  must  have  with  respect  to  the  other  two,  the 
same  magnitude  as  the  diagonal  AD  has  with  respect  to  the  two 
lateral  lines  AB  and  AC.  From  this  proposition  it  will  be  easily  per- 
ceived in  what  manner  a  single  force  may  be  found  equivalent  to  any 
number  of  given  forces. 


24  Solid  Bodies. 

Equilibrium  of  Bodies  which  move  about  a  Fixed  Axis. 

85.  Let  LM  (fg.  4.)  be  a  body  without  gravity,  turning  about  a 
fixed  point  C  to  which  it  is  attached.  If  this  body  is  acted  upon  by  a 
single  force  AC,  the  direction  of  which  passes  through  the  point  C, 
an  equilibrium  takes  place ;  whether  this  force  tends  toward  the  point 
C  or  in  an  opposite  direction.     For  in  these  two  cases,  its  effect  is 
equally  destroyed  by  the  resistance  of  the  fixed  point.     But  every 
force,  the  direction  of  which  does  not  pass  through  this  point,  will 
produce  a  motion  of  rotation. 

86.  Let  there  be  two  forces  P  and  Q,  applied  at  another  point  A 
of  the  body,  and  acting  in  the  directions  AP  and  AQ.     Let  us  sup- 
pose also  that  the  plane  of  the  two  lines  AP,  AQ,  contains  the  fixed 
point  C.     At  the  point  C  let  the  parallelogram  ABCD  be  formed. 
It  will  be  easily  seen,  that  there  will  then  be  an  equilibrium,  if  we 
have  the  proportion  P  :  Q  :  :  AB  :  AD. 

87.  If  we  draw  from  the  point  E,  the  lines  CE  and  CF,  perpen- 
dicular to  the  directions  of  the  forces,  it  may  be  demonstrated  by 
geometry,  that  AB  :  AD  :  :  CE  :  CF.     Consequently, 

P  :  q  :  :  CE  :  CF* 

88.  There  would  also  be  an  equilibrium  if  the  two  forces  'P,  Q, 
applied  at  the  point  A,  were  in  other  directions,  for  example,  towards 
G7  and  H' ;  provided  that  their  ratio  remained  the  same,  and  their 
plane  passed  always  through  the  fixed  point  C.     Indeed,  if  these 
conditions  were  fulfilled,  it  would  be  immaterial  whether  the  point  A 
were  placed  within  or  without  the  body  LM ;  and  if  A  were  with- 
out the  body,  the  distance  at  which  it  might  be  situated  would  make 
no  difference.     It  might,  therefore,  be  at  an  infinite  distance  ;  that  is, 
AP  and  AQ  might  be  parallel. 

In  all  cases,  without  exception,  there  is  an  equilibrium,  when 
P  :  Q  :  :  CE  :  CF,  provided  the  plane  of  the  two  forces  contains 
the  fixed  point. 

89.  From  this  proportion,  taking  the  product  of  the  means  and 
extremes,  we  have  P  .  CF  =  Q .  CE. 

*  In  the  triangles  BCF,  DCE,  F  and  E  are  right  angles,  and  the 
angle  B  is  equal  to  D,  each  being  equal  to  BAD ;  consequently,  these 
triangles  are  similar,  and  we  have  DC  :  BC  :  :  CE  :  CF ;  whence 
AB:AD::CE:  CF. 


Centre  of  Gravity  of  Bodies.  25 

The  perpendiculars  CE,  CF,  are  called  the  distances  of  the 
forces  ;  and  the  product  of  a  force  by  its  distance,  is  called  the  stati- 
cal moment  or  simply  the  moment  of  this  force.  Accordingly,  we 
may  express  the  condition  of  equilibrium  in  the  following  manner. 
Two  forces,  Pand  Q,  wiU  keep  a  body,  moveable  about  a  fixed  point 
C,  in  equilibrium,  when  their  moments  are  equal.* 

90.  If  more  than  two  forces  act  upon  the  body  LM,  we  consider 
separately  the  forces  which  tend  to  turn  the  body  one  way,  and  those 
which  tend  to  turn  it  the  other.  An  equilibrium  takes  place  when 
the  sum  of  the  moments  of  the  forces  which  tend  to  turn  the  body  in 
one  direction,  is  equal  to  the  sum  of  the  moments  of  tfte  forces  which 
tend  to  turn  it  in  the  opposite  direction. 

On  these  propositions  (85 — 90)  is  founded  the  entire  theory  of 
pullies,  wheels,  and  levers.  But  its  detailed  exposition  belongs  to 
the  science  of  machines. 


CHAPTER  XI. 

Centre  of  Gravity  of  Bodies. 

91.  THE  gravity  of  each  of  the  particles  of  a  solid  body  may  be 
considered  as  a  force  which  acts  upon  it  in  a  vertical  direction.  If 
we  attempt  to  place  a  body  JIB  (fig.  5)  upon  a  support  CD,  which 
has  a  sharp  point,  it  is  evident,  from  what  has  just  been  said  of  equi- 
librium, that  there  must  be  in  the  body  a  point,  which  being  support- 
ed, the  whole  body  will  be  kept  in  equilibrium. 

This  conclusion  is  always  correct,  in  whatever  position  the  body 
is  placed  on  the  support.  If  we  make  this  experiment  in  three  direc- 
tions perpendicular  to  each  other,  we  shall  determine  the  only  point, 

*  If  the  body  actually  moved  about  the  point  C,  the  velocities  of 
rotation  of  the  points  F  and  E  would  be  as  FC  to  EC.  If  we  call 
V  the  velocity  of  the  point  F,  and  V  the  velocity  of  the  point  E,  we 
shall  have  V :  V  :  :  FC  :  EC.  Consequently,  P  :  Q  :  :  V  :  V  and 
P  X  V=  Q  X  V'.  The  product  PV,  QF',  of  each  force  by  its 
virtual  velocity,  is  called  the  mechanical  moment  of  this  force.  Ac- 
cordingly, we  say  there  will  be  an  equilibrium,  when  the  mechanical 
moments  of  the  forces  P  and  Q  are  equal. 

Elem.  4 


26  Solid  Bodies. 

which  being  sustained,  the  whole  body  will  be  kept  in  equilibrium. 
This  point  is  called  the  centre  of  gravity  of  the  body. 

92  When  the  centre  of  gravity  is  sustained,  the  whole  weight  ol 
the  body  is  sustained,  in  the  same  manner  as  if  it  were  all  concen- 
trated in  this  single  point.  This  supposition  is  true  with  respect  to 
equilibrium,  but  it  is  not  always  applicable  to  a  state  of  motion. 

93.  We  must  not  confound  the  centre  of  gravity  with  the  centre 
of  figure.     They  coincide  only  in  bodies  of  a  uniform  density  ;  bi 
in  bodies  of  unequal  density  the  centre  of  gravity  is  always  nearer 
the  denser  part;  it  is  even  true  that  the  centre  of  gravity  is  D 
always  in  the  interior  of  the  body.     In  rings,  for  example,  ai 
many  other  figures,  it  is  exterior  to  the  body. 

94.  The  centre  of  gravity  may  be  sustained  in  two  ways ;  from 
above,  if  the  body  is  suspended ;  or  from  beneath,  if  it  is  placed  upon 
some  support.     When  a  body  is  suspended  by  a  thread,  the  centre 
of  gravity  is  always  in  the  direction  of  the  thread  produced.     In  this 
manner  we  can  find  the  centre  of  gravity  of  a  body  more  readily, 
than  by  the  method  above  given.     We  have  only  to  attach  a  thread 
to  two  points  in  the  body,  and  suspend  it  successively  in  the  two 
corresponding  directions. 

95.  When  a  body  is  placed  upon  a  support,  its  situation  is  the  less 
secure,  according  as  the  surface  sustained  is  smaller,  and  the  centre 
of  gravity  more  distant  from  the  middle  of  this  surface. 

96.  When  the  form  of  a  body  is  variable,  as  is  the  case  with  the 
bodies  of  men  and  animals,  the  centre  of  gravity  is  also  variable, 
When  a  man  stands  erect  and  lets  his  hands  fall  equally  on  each 
side,  the  centre  of  gravity  is  in  the  lower  part  of  the  abdomen, 
nearly  between  the  hips.     It  may  be  inferred  from  what  precedes, 
that  the  position^pf  this  point  is  of  the  utmost  importance  as  it  re- 
spects the  stability  of  the  body ;  and  hence  the  almost  involuntary 
efforts  which  are  made  to  keep  it  in  its  place  and  thus  prevent  a  fall. 

97.  There  are  many  physical  phenomena  the  explanation  of 
which  depends  upon  the  centre  of  gravity.  As  they  are  instructive, 
they  merit  some  attention.  Of  this  description  are  the  cylinder  rolling 
up  an  inclined  plane  ;  the  double  cone  which  seems  to  rise  against 
the  effort  of  gravity ;  the  little  vaulter,  &tc.  The  difficult  art  of 
rope-dancing  refers  itself  also  to  this  theory.  But  what  is  of  par- 
ticular importance,  is  the  application  of  the  centre  of  gravity  to  the 
theory  of  balances. — [See  Cam.  Mech.  p.  161.] 


Laws  of  uniformly  accelerated  Motion.  27 


CHAPTER  XJI. 

Free  Descent  of  Heavy  Bodies. — Laws  of  Motion  uniformly  accele- 
rated. 

98.  WHEN  a  body  falls  freely  by  the  mere  action  of  terrestrial 
gravity,  its  velocity  increases  at  each  instant,  because  gravity  acts 
upon  it  constantly  during  its  descent;  but  as  gravity  at  the  same 
place,  and  for  small  distances  from  the  surface  of  the  earth,  may  be 
considered  as  an  invariable  force,  the  velocity  of  a  falling  body  must 
increase  just  as  much  in  one  instant  as  in  another.     Hence  we  may 
say  that  a  falling  body  has  a  motion  uniformly  accelerated. 

99.  The  descent  of  a  body  is  much  too  rapid  to  permit  the  laws 
of  its  motion  to  be  ascertained  by  direct  observation.     But  the 
machine  invented  by  Atwood*  furnishes  a  convenient  method  of  re- 
tarding the  descent,  so  that  without  altering  the  essential  laws  of  the 
motion,  we  can  observe  its  phenomena  from  one  second  to  another. 

100.  By  means  of  this  machine  we  learn, 

(1.)  That  in  every  uniformly  accelerated  motion  the  spaces  de- 
scribed are  as  the  squares  of  the  times.  If,  therefore,  we  call  g  the 
space  described  during  the  first  second,  however  great  or  small  it 
may  be,  and  if  we  designate  by  S  the  space  described  in  T  seconds, 
we  shall  have  S=gT2. 

(2.)  That  the  velocities  are  to  each  other  as  the  times,  and  that  the 
velocity  which  the  body  has  at  the  end  of  each  second,  is  found  by 
multiplying  double  the  space  described  during  the  Jirst  second,  by  the 
time.  If  then  we  call  V  the  velocity  which  the  body  acquires  in  T 
seconds,  we  have  V  =.  2g  T. 

(3.)  From  these  two  propositions  we  deduce  the  following  ;  The 
spaces  described  are  as  the  squares  of  the  velocities.^ 

*  [A  description  of  Atwood's  machine  may  be  found  in  Adams's 
Lectures  on  Natural  Philosophy,  vol.  iii.  p.  125.] 

t  Let  S  be  the  space  passed  over  by  a  heavy  body  during  the  time 
T,  in  virtue  of  the  action  of  gravity  ;  let  us  conceive  the  time  T,  to 
be  divided  into  a  certain  number  n,  of  intervals  <,  equal  among  them- 
selves ;  whence  we  shall  have  T  =  n  t.  We  can  compare  the  mo- 
tion sought  with  that  of  a  body  without  weight  which  receives  at  the 
end  of  each  of  the  instants  t}  2  f,  3  t,  an  impulse  capable  of  making 


28  Solid  Bodies. 

101.  In  order  to  express,  by  numbers,  all  the  circumstances  of  a 
motion  uniformly  accelerated,  it  is  only  necessary  to  knowg  ;  that 
is,  the  space  described  in  the  first  second.  It  is  ascertained  by 
means  of  the  apparatus  above  referred  to,  that  when  a  body  falls 

__  _  _  __  .  _ 

it  describe  uniformly  the  space  v  in  a  unit  of  time  ;  this  comparison 
will  approach  the  nearer  to  the  truth  as  the  internals  t,  which  separate 
the  successive  impulses  of  the  force  become  smaller,  and  finally  the 
error  will  disappear  entirely  in  the  results  which  are  independent  of 
the  absolute  value  of  these  instants.  Let  us  follow  then  the  conse- 
quences of  our  supposition,  and  analyze  the  effects  which  the  succes- 
sive impulses  will  have  produced  at  the  end  of  the  time  T  upon  the 
moving  body. 

The  instant  0  is  the  time  of  commencement.  The  moveable  body 
then  receives  an  impulse  capable  of  making  it  describe  the  space  v  in 
a  unit  of  time.  This  first  impulse  acts  upon  it  during  the  time  T;  it 
causes  it  then  to  Describe,  in  this  interval,  the  space  TV* 

At  the  instant  following  t  the  moving  body  receives  an  additional 
impulse  equal  to  the  preceding,  but  this  impulse  acts  only  during  the 
time  T  —  t  ;  it  causes  it  then  to  describe  the  space  (T  —  t)  v. 

By  examining,  in  this  way,  the  spaces  which  the  successive  impulses 
cause  the  body  to  describe,  to  the  end  of  the  time  T,  it  will  be  seen 
that  these  spaces  form  the  decreasing  arithmetical  progression, 

TV;  (T—t)v;  (  T  —  2  t)  v  ;....*  y. 

Then  the  number  of  terms  is  equal  to  the  number  of  instants  t,  that 
is,  equal  to  n.  By  substituting  for  T  its  value  n  t,  this  progression 
will  become 

ntv,  (n-i)tv;  (n  —  2)  *  v  ;....*  v. 

The  sum  of  these  partial  spaces  is  the  whole  space  really  passed 
over  by  the  moving  body.  Now  this  sum  is 


X      - 


2  2 

We  have  represented  this  space  by  S ;  we  have  then 

g_nx(«+l)  X  tv 

2  ' 

t 

or  substituting  for  »  its  value    T, 
r.aii«i;  '! 


Laws  of  uniformly  accelerated  Motion.  29 

freely  and  without  obstruction,  it  describes  16,1  feet  in  the  first 
second,  and  acquires  a  velocity  equal  to  32,2.  This  being  known,  we 
can  easily  calculate  the  space  described  and  the  velocity  acquired, 
after  any  determinate  time ;  and  generally,  knowing  one  of  these 
three  things,  the  time,  space,  or  velocity,  we  can  find  the  other  two. 


Let  us  designate  by  g  the  space  passed  over,  in  this  manner,  by  the 
moving  body  in  a  unit  of  time,  and  suppose  T  commensurable  with 
this  unit.  We  can,  in  like  manner,  divide  the  unit  of  time  into  a 
number  n1  of  intervals  equal  among  themselves  and  each  equal  to  £; 
and  by  resuming  the  reasoning,  in  the  same  manner,  for  this  second 
case,  we  shall  find 

g  =  -^ 

whence  we  deduce 


g  l  +  « 

or,  what  amounts  to  the  same  thing, 

S-T.      T(T-i)t 

g~          l  +  « 

The  smaller  t  becomes,  the  more  the  factor  —  -j—  -  will  be  dimin- 

01 

ished,  and  consequently  the  nearer  the  ratio  —  will  approach  to  an 

to 

equality  with  T2.  Now,  by  supposition,  t  is  the  interval  of  time  which 
elapses  between  the  successive  impulses  of  the  force,  and  this  inter- 
val is  entirely  imperceptible  to  our  senses,  in  the  case  of  gravity. 
So  that  we  must  here  suppose  t  —  0,  which  gives 


that  is,  in  uniformly  accelerated  motion,  the  spaces  passed  over  are 
proportional  to  the  squares  of  the  times. 

Net  us  now  examine  the  value  of  the  acquired  velocity  after  the 
time  T.  Let  V  be  this  velocity  ;  it  will  be  equal  to  the  sum  of  all 
the  impulses  given,  to  the  moving  body,  during  the  time  T.  Now, 
since  T=  nt,  it  is  clear  that  the  Moving  body  receives  a  number  n 
of  impulses  in  this  interval  ;  and  as  each  one  of  them  gives  to  it  the 

velocity  r,  their  sum  will  be  n  o  or  T  X  ^  ;  we  shall  then  have 

r=  x  rf 


30  Solid  Bodies. 

We  can,  in  like  manner,  determine  all  the  circumstances  of  every 
other  motion  uniformly  accelerated,  when  we  know  the  correspond- 
ing value  of  g.  This  value,  therefore,  is  the  measure  of  all  motions 
of  this  kind  ;  and  is  hence  called  the  measure  of  acceleration,  or  the 
accelerating  fo  rce  .  * 

Now,  by  calling  g  the  space  passed  over  in  a  unit  of  time,  we  have 
seen  that 

0  +  0  »  . 

^  =  —  —  «' 

dividing  these  equations,  member  by  member,  they  become 
V        2T  r_~v       2Tt 


The  more  t  diminishes,  the  nearer  the  ratio  —  approaches  to  an 

o 

equality  with  2  T;  in  the  case  of  gravity  we  must  make  t  =  0,  and 
we  have  rigorously 


that  is,  the  velocity  is  proportional  to  the  time. 

We  have  supposed  that  the  time  T  is  commensurable  with  the  unit 
of  time.  If  this  condition  be  not  fulfilled,  we  should  render  it  exact 
by  adding  to  the  time  T,  a  portion  of  the  interval  t ;  and  then  we 
should  commit  an  error  the  extent  of  which  would  depend  on  the 
space  described  by  the  moving  body  during  this  small  interval.  But, 
as  we  afterward  make  t  nothing,  it  is  evident  that  the  error  will,  in 
like  manner,  become  nothing.  This  shows  that  the  preceding  re- 
sults are  independent  of  the  hypothesis  of  the  commensurability 
which  has  helped  us  to  discover  them. 

*  The  two  formulas,  developed  in  the  preceding  note,  V=  2  g  T, 
and  S  =.  g  T2,  are  the  fundamental  propositions  by  which  all  ques- 
tions in  uniformly  accelerated  motion  are  solved.  By  supposing  that 
g  is  given,  it  will  be  readily  perceived  that  by  means  of  it,  whenever 
one  of  these  quantities  T,  S,  V,  is  given,  the  two  others  can  be  de- 
termined ;  for  if  it  is  T  which  is  given,  the  two  preceding  equations 
immediately  give  V  and  S ;  if  V  is  given,  we  deduce 
__  V  fra 

and  lastly,  if  S  is  given,  from  the  equation  S  =  g  T8,  we  obtain 


Laws  of  uniformly  accelerated  Motion.  31 

102.  When  a  body  is  projected  upwards  in  a  vertical  direction, 
it  is  clear  that  gravity  uniformly  diminishes  the  velocity,  in  precisely 
the  same  ratio  as  it  increases  the  velocity  on  its  descent.   The  body, 
therefore,  ascends  until  the  continued  action  of  gravity  has  destroyed 
all  the  velocity  communicated  by  the  first  impulse.     Then  it  de- 
scends, and  it  is  plain  that  it  must  recover  at  each  point  of  its  de- 
scent the  same   velocity  it  had  at  this  same   point   in  ascending. 
This  remark  enables  us  to  determine  the  height  which  a  body  will 
attain,  when  we  know  the  velocity  with  which  it  is  projected.* 

103.  The  foregoing  propositions  are  only  rigorously  true  when 
bodies  fall  in  a  space  void  of  air.     In  the  air,  on  the  contrary,  their 
cannot  be  an  exactly  uniform  acceleration,  because  the  body,  at  each 
instant  must  displace  a  quantity  of  air,  and  must  lose  all  the  motion 
which  it  communicates  to  the  air.    If,  therefore,  a  falling  body  has  a 
small  mass  and  large  bulk,  its  motion  must  evidently  b«  very  much 
retarded  by  the  resistance  of  the  air  ;  and  this  is  conlormable  to  ex- 
perience.    Reciprocally  the  resistance  of  the  air  is  less  sensible  in 
proportion  as  the  falling  body  has  a  large  mass  and  small  bulk. 


T=         ,  and  hence  V 


=  2g  T=  2g     II  = 
V8 


If  the  question  is  concerning  the  free  descent  of  bodies  produced 
by  the  action  of  gravity,  it  is  only  necessary  to  substitute  for  g  16,1 
feet.  If  the  inquiry  relates  to  any  other  uniformly  accelerated  mo- 
tion we  must  substitute  for  g  the  value  which  belongs  to  it. 

*  Let  U  be  the  velocity  of  projection  given  to  a  body  at  the  mo- 
ment it  is  propelled  upward,  and  count  the  time  T  from  this  epoch. 
The  body  will  cease  to  ascend  when  the  repeated  impulses  of 
gravity  to  make  it  descend  have  amounted  to  the  sum  of  the  velo- 
cities or  a  total  velocity  V  equal  to  17.  Now  designating  by  T 
the  time  after  which  this  will  happen,  we  shall  have  V=2gT, 
according  to  the  laws  of  gravity  ;  it  is  necessary  then  that  this  pro- 

duct 2  g  T  should  be  equal  to  17,  which  gives  T  =.  —  ;  this  is  tht 

time  at  which  the  moving  body  stops  ;  afterwards  it  begins  to  descend. 
When  in  the  space  S  which  it  will  then   have  described,  it  will  be 

f/2 

found  equal  to  g  T2  or  —  ;  an  expression  which  may  be  easily  re- 
duced to  numbers  when  the  initial  velocity  U  is  given. 


Solid  Bodies. 


Motion  down  Inclined  Planes. 

104.  If  a  heavy  body  D  (fig.  6)  be  placed  upon  au  inclined 
plane  AB,  it  rolls  or  slides  toward  the  bottom,  but  with  less  force 
than  it  would  have  in  a  free  vertical  descent.     The  magnitude  of 
this  force  may  be  determined  in  the  following  manner.     Let  D  be 
the  centre  of  gravity  of  the  body,  and  draw  from  the  centre  D  to 
the  inclined  plane,  the  vertical  DE,  representing  the  absolute  force 
of  gravity,  or  the  weight  of  the  body  ;  that  is,  the  force  that  would 
be  exerted  in  the  case  of  a  free  descent.     Then  draw  the  two  lines 
DF,  DG,  the  first  perpendicular  and  the  second  parallel  to  AB. 
The  force  DE  may  be  considered  as  decomposed  into  two  others, 
one,  DF,  perpendicular,  the  other,  DG,  parallel  to  AB.     The  first 
can  produce  no  motion,  being  counteracted  by  the  resistance  of  the 
plane. 

The  second,  DG,  on  the  contrary,  acts  in  the  direction  in  which 
the  body  moves,  and  produces  all  the  motion  it  has.  The  vertical 
force,  therefore,  that  is,  the  weight  of  the  body,  is  to  the  force  with 
which  it  moves  down  the  inclined  plane,  as  DE  is  to  DG.  If  from 
the  points  A  and  B,  taken  at  pleasure,  on  the  inclined  surface,  we 
draw  the  vertical  line  AC  and  the  horizontal  line  BC,  we  have  the 
triangle  ABC,  similar  to  the  triangle  DEG ;  and 

DE:DG::AB:AC; 

that  is,  the  vertical  force  is  to  the  oblique  force  as  the  length  of  the 
inclined  plane  is  to  its  height.  There  is  a  particular  mstrument  called 
the  inclined  plane,  intended  to  verify  this  proportion. 

105.  At  whatever  point  of  the  inclined  surface  the  body  is  situated, 
the  accelerating  force  will  always  be  the  same  ;  this  motion,  there- 
fore, is  uniformly  accelerated.     In  the  case  of  free  vertical  descent, 
we  have  called  this  force  g,  which  is  equal  to  16,1  feet;  and  the 

AC9 

proportion  AB  :  AC  :  :g  :  ~g,  enables  us  to  determine  all  the 
circumstances  of  motion  down  an  inclined  plane.* 

*  Galileo,  who  first  developed  accurately  the  laws  which  govern 
the  motion  of  falling  bodies,  made  use  of  the  inclined  plane  to  verify 
the  result. 


Motion  doum  Inclined  Planes.  83 

106.  One  circumstance,  very  important  in  its  consequences,  is, 
that  the  body  in  descending  from  A  to  B,  acquires  exactly  the  same 
velocity  as  if  it  had  fallen  freely  from  A  to  C  ;  for  the  velocities  at 
each  point  are  to  each  other  aswlB  to  AC,  that  is,  as  the  lengths  of 
the  planes.* 

107.  If  we  suppose  two  bodies,  of  which  one  falls  vertically  from 
A  to  C,  and  the  other  obliquely  from  A  to  B,  but  both  having  in  A  the 
same  determinate  velocity,  it  is  clear  that  they  will  also  have  the 
same  velocity  in  C  and  B.     Let  ABCD  (Jig.  7)  then  be  a  plane, 
interrupted  at  pleasure   by   the   angles  B,    C,   &c.     If  we   draw 
through  A  and  D  the  vertical  lines  JlKand  DE,  and  through  A, 
B,  C,  D,  the  horizontal  lines  AE,  FG,  HI,  KD ;  it  is  evident 
that  a  body  sliding  down  the  broken  plane  will  have  at  A,  B,  G, 
the  same  velocities  as  the  body  falling  vertically  from  A  will  have  at 
Ft  H,  K,  or  one  from  E,  in  G,  /,  D. 

108.  As  the  number,  magnitude,  and  situation  of  the  interruptions, 
are  perfectly  arbitrary,  this  proposition  is  likewise  true  of  a  curved 
line  AB  (fig.  8).     If  we  draw  at  pleasure  the  horizontal  line  AC 
and  the  vertical  line  CB,  we  can  determine  the  velocity  of  a  body 
moving  along  AB,  at  any  points  D,  F,  by  drawing  through  these 
points  the  horizontal  lines  DE  and  FG.     It  will  be  exactly  the  same 
as  if  the  body  had  fallen  freely  from  C  to  jP  or  G.     Hence  we  de- 
duce the  following  remarkable  proposition ;    When  a  body  passes 

from  one  inclined  surface  to  another,  it  acquires  the  same  velocity, 
whatever  course  it  takes.  In  reality,  however,  the  resistance  of  the 
air  may  cause  a  difference. 


*  If  we  represent  by  g  the  accelerating  force  belonging  to  a  verti- 
cal descent,  that  which  urges  a  body  down  an  inclined  plane  will  be 
found  by  the  proportion 


Let  V  be  the  velocity  acquired  by  a  body  descending  freely  from  A 
to  C,  and  V  the  velocity  belonging  to  an  oblique  fall  from  A  to  B, 
we  shall  have,  according  to  note  to  article  101, 


V  =  2  VFX^C",  and  V  =  2  g  X  AB  = 


that  is,  exactly  equal  to  the  preceding. 
Elem.  5 


34  Solid  Bodies. 

CHAPTER  XIII. 

Free  Curvilinear  Motion. 

(1.)  Projectiles. 

109.  WHEN  a  heavy  body  receives  an  impulse  in  a  direction  ob- 
lique to  the  vertical,  it  describes  a  curve,  the  form  of  which  may  be 
found  in  the  following  manner ;  w- 

From  the  point  A  (fig.  9),  where  the  motion  commences,  draw 
the  vertical  line  A  25  ;  take  A  1  for  unity  ;  then  beginning  from  A, 
set  off,  according  to  this  unity,  the  distances  4,  9,  16,  25,  according 
to  the  progression  of  square  numbers.  If  then  A  1  represents  the 
space  described  by  a  heavy  body  in  the  first  second  of  descent,  it  is 
evident  that  the  points  4,  9,  16,  25,  are  the  places  at  which  it  will 
be  found,  at  the  end  of  the  2d,  3d,  4th,  5th,  seconds.  At  the  in- 
stant when  the  body  begins  to  fall,  let  us  suppose  that  it  receives  an 
impulse  in  the  direction  AF,  such  that  if  the  body  had  no  weight 
and  yielded  entirely  to  this  impulse,  it  would  describe  AB  in  the  first 
second,  and  in  the  following  seconds,  the  spaces  BC,  CD,  DE, 
EF,  equal  to  the  first. 

Now  the  only  case  in  which  a  body  can  have  two  motions  at  once, 
is  when  it  moves  in  a  space  which  is  also  in  motion.  In  the  case 
before  us,  the  motion  produced  by  gravity  takes  place  in  the  line 
A  25,  and  to  represent  the  motion  communicated  by  the  impulse,  we 
must  suppose  this  line  to  move  together  with  the  body  in  a  direction 
always  parallel  to  itself,  and  with  a  velocity  equal  to  the  velocity  of 
projection.  Accordingly,  at  the  end  of  the  2d,  3d,  4th,  5th  seconds, 
the  line  A  25  will  occupy  the  positions  B  p,  C  Y,  D  S,  E  f ,  F  </> .  If 
then  we  draw  1  6,  4c,  9  d,  16  e,  25/,  parallel  to  AF,  it  is  evident 
that  the  body  will  be  in  b,  at  the  end  of  the  first  second,  in  c  at  the 
end  of  the  2d,  and  so  on.  And  if  these  points  are  connected  by  the 
curve  line  A  b  c  d  e/,  this  line  will  represent  the  path  of  the  body. 

110.  It  may  be  shown  thst  this  curve  belongs  to  the  species  called 
Paral)olas.-[See   Cam.  Math.  Trig.  art.  172 ;  also  Cam.  Mech. 
art.  303.] 

111.  Here  experiment  differs  very  widely  from  theory,  on  ac- 
count of  the  resistance  of  the  air ;  and  although  it  would  be  very 
difficult  to  calculate  this  resistance,  it  is  obvious  that  its  effect  would 


Central  Forces.  3& 

be  to  make  the  part  G?/,  which  theoretically  has  the  same  curvature 
as  the  part  AG,  increase  very  rapidly  in  its  curvature,  as  we  remove 
from  the  point  A  of  departure. 


(2.)   Central  Forces. 

1 12.  From  what  precedes,  we  may  conclude  that  in  order  to  pro- 
duce a  curvilinear  motion,  at  least  two  forces  are  requisite  ;  and  that 
one  at  least  of  these  must  be  an  accelerating  force,  while  the  other 
may  be  an  instantaneous  impulse.     Among  the  endless  variety  of 
cases  of  this  kind,  there  is  no  one  more  interesting  to  the  student  of 
natural  science,  than  that  in  which  a  force  continually  draws  a  body 
towards  a  certain  centre,  while  the  body  has  received  an  exterior 
impulse  from  another  force.     These  combinations  of  forces  are  call- 
ed central  forces,  and  the  motions  which  they  produce  central  mo- 
tions. 

113.  Let  C  (jig.  10)  be  the  point  towards  which  a  body  situated 
in  A  is  constantly  urged,  but  let  us  suppose  that  the  body  receives  at 
the  same  time  a  motion  in  the  direction  AD.     Although  the  central 
force  acts  constantly  and  without  interruption,  we  shall  for  a  moment 
suppose,  for  the  convenience  of  illustration,  that  it  acts  by  separate 
impulses,  and  that  it  repeats  its  action  in  very  small  but  equal  por- 
tions of  time,  which  we  call  instants.    In  the  first  instant,  the  central 
force  tends  to  move   the  body  from  A  to  /?,  and  the  lateral  force 
from  A  to  IX;  it  will  therefore  describe  the  diagonal  AE  of  the  par- 
allelogram ABDE.     Produce  AE  till  EG  =  AE,  and  EG  will  be 
the  path  which  would  be  described  in  the  second  instant,  if  the  body 
were  free  ;  but  at  the  beginning  of  this  instant,  the  central  force  will 
tend  to  move  it  from  E  to  F ;  it  will,  therefore,  describe  the  diago- 
nal EH,  and  so  on.     It  is  obvious  that,  according  to  the  preceding 
supposition,  the  body  would  describe  a  broken  line  AEHL.     But 
as  the  central  force  does  not  act  by  separate  impulses,  it  is  certain 
that  the  true  path  of  the  body  will  be  a  curved  line,  the  form  of 
which  may  be  infinitely  varied,  either  by  the  difference  of  intensity 
and  direction  of  the  lateral  force,  or  by  difference  of  the  power  and 
the  laws  of  the  central  force. 

114.  Many  writers  call  the  force  which  urges  the  body  towards 
C,  a  centripetal  force,  to  which  a  contrary  force  is  opposed,  called 
centrifugal.     But  some  confusion  exists  as  to  what  constitutes  this 


3(5  Solid  Bodies. 

last;  and  a  more  precise  limitation  of  the  term  is  rendered  neces- 
sary ;  especially  as  the  terms  centripetal  and  centrifugal  are  in  very 
common  use.  Huygens,  first  used  the  term  centrifugal  force,  and 
the  meaning  he  affixed  to  it  we  shall  now  explain.  The  action  of 
the  forces  by  which  the  body  in  question  is  made  to  describe  the 
diagonal  AE,  may  be  represented  in  a  manner  somewhat  different 
from  that  above  stated.  Let  BA  be  produced  till  JIM  =  AB,  and 
draw  DM;  AEDM  is  a  parallelogram,  and  we  may  consider  the 
force  AD  as  decomposed  into  two  others  AM,  AE,  the  first  of 
which,  AM,  is  equal  and  opposite  to  the  centripetal  force  ;  these  two 
forces,  therefore,  destroy  each  other,  and  the  body  can  only  be 
acted  upon  by  the  force  DM  or  its  equal  AE.  This  force  AM  is 
what  Huygens  calls  the  centrifugal  force  ;  it  will  be  readily  seen, 
that  in  all  cases,  it  is  equal  and  opposite  to  the  centripetal  force.  In- 
stead of  using  the  term  in  this  sense,  many  call  the  lateral  force  AD 
the  centrifugal  force ;  but  it  would  be  better  to  call  it  the  tangential 
force,  since  it  always  tends  to  impel  the  body  in  the  direction  of  a 
tangent  to  the  curve  which  it  describes. 

115.  It  is  by  the  effect  of  the  centrifugal  force  that  the  strings  of 
a  sling  are  felt  to  be  stretched  when  we  whirl  it ;  and  it  is  by  the 
effect  of  the  tangential  force,  that  the  stone  flies  off  in  the  direction 
of  a  tangent,  when  let  loose  from  the  sling.  It  is  also  by  the  effect 
of  the  tangential  force,  that  water  dashes  over  the  sides  of  a  vessel, 
when  made  to  revolve  rapidly.  An  ingenious  machine  has  been 
invented  called  the  whirling  table,  for  the  purpose  of  illustrating  all 
the  laws  of  central  motion.  A  description  of  it  may  be  found  in 
[EnfeUFs  Institutes,  &,c.  and  Adams's  Lectures  on  Nat.  Phil.'} 

A  ball  suspended  by  a  thread  furnishes  a  very  easy  and  simple 
method  of  observing  the  formation  of  central  motions.  As  this  ball 
can  only  be  at  rest  when  situated  in  a  vertical  line  below  the  point 
of  suspension,  it  always  returns  to  this  position  when  made  to  diverge 
from  it  j  and  thus  the  effect  of  gravity  here  represents  a  central 
force.  If  we  give  the  ball  an  impulse  in  a  direction  oblique  to 
the  vertical,  it  takes  a  curvilinear  motion  about  the  central  point. 
This  motion  may  be  elliptical  or  circular,  according  to  the  difference 
of  direction  and  force  of  the  impulse. 

H6.  It  is  upon  this  simple  action  of  central  forces,  that  the  won- 
derful motions  of  the  heavenly  bodies  depend.  Kepler  first  dis- 
covered the  principal  laws  of  these  motions  ;  but  he  did  not  deduce 
them  from  the  laws  of  mechanics  ;  he  ascertained  them  only  by 


Curvilinear  Motions.  37 

great  industry  and  acuteness  of  observation.  Newton  laid  the  foun- 
dation of  the  theoiy  of  these  motions,  and  discovered  the  simple 
laws  of  universal  gravitation,  by  which  the  harmony  of  all  these 
bodies  is  preserved  and  perpetuated. 


CHAPTER  XIV. 

Motion  in  given  Lines. 

117.  A  BODY  may  in  various  ways  be  made  to  take  a  different 
motion  from  that  which  it  would  have  taken,  by  the  free  action  of  the 
forces  exerted  upon  it.     Hence  in  mechanics  we  distinguish  between 
&free  motion  and  a  motion  in  given  lines.     We  shall  here  examine 
only  two  of  the  latter  kind  of  motions  ;  namely,  curvilinear  motions, 
and  the  oscillation  of  a  pendulum. 

(1.)  Curvilinear  Motions. 

118.  When  a  free  solid  body  receives  an  impulse  which  does  not 
pass  through  its  centre  of  gravity,  it  takes  a  motion  compounded  of 
two  others.    1.  A  uniform  motion  of  translation  in  space,  common 
to  all  its  particles.     2.  A  motion  of  rotation,  also  uniform,  about  an 
axis  passing  through  the  centre  of  gravity,  but  the  direction  of  which 
may  be  variable  or  constant  in  the  interior  of  the  body.   In  all  bodies 
we  may  draw  through  the  centre  of  gravity  three  lines  at  right  an- 
gles to  each  other,  which  are  permanent  axes  of  rotation  ;  that  is,  if 
the  rotation  has  once  commenced  about  one  of  these  axes,  it  will 
always  continue  to  take  place  about  this  same  axis,  provided  that  the 
body  experiences  no  resistance  or  impulse  to  disturb  the  freedom 
which  we  have  supposed  in  its  motions.     This  appears  to  be  the 
case  with  the  heavenly  bodies.     All  these  results  are  demonstrated 
in  treatises  on  mechanics. 

119.  Let  us  consider  in  particular  the  case  of  a  uniform  motion 
of  rotation  about  a  permanent  axis,  passing  through  the  centre  of 
gravity.     Let  this  centre  be  represented  by  C  (fg.  llj,  and  let  the 
axis  of  rotation  be  perpendicular  to  the  plane  of  the  figure.     In  this 
case  every  point  of  the  body  will  describe  about  the  axis  of  rotation 


4C45SG 


38  Solid  Bodies. 

the  circumference  of  a  circle  ;  the  point  A,  for  example,  will  des- 
cribe the  circumference  AEKD.  During  this  motion  of  rotation, 
the  force  of  cohesion,  by  which  the  particles  of  the  body  are  united, 
will  perform  the  part  of  a  centripetal  force.  But  however  great 
may  be  its  effect,  considered  in  this  relation,  it  does  not  depend  at 
all  upon  its  proper  intensity,  but  simply  upon  the  motion  and  form 
of  the  body  moved.  To  be  easily  convinced  of  this,  let  AG  be  the 
arc  which  the  point  A  describes  in  an  infinitely  small  interval  of  time. 
Draw  the  tangent  AB,  and  complete  the  parallelogram  AFGH. 
Now,  if  AF  represents  the  force  with  which  A  tends  to  follow  its 
tangential  motion  AF,  AH  is  the  central  force  by  which  it  is  drawn 
towards  C ;  and  this  evidently  does  not  depend  at  all  upon  the  physi- 
cal properties  of  the  body,  but  upon  the  velocity  of  the  point  A,  and 
its  distance  from  the  axis  of  rotation  C.  Thus,  in  the  rotation  of  a 
solid  body,  there  must  always  be  produced  at  the  surface  of  the 
body  a  centripetal  force,  which  is  necessary  to  prevent  the  particles 
from  escaping.  Such  a  force  must  also  exist  in  each  point  of  a 
revolving  body,  and  it  is  the  more  intense  according  as  this  point 
is  farther  distant  from  the  axis  of  rotation  C,  that  is,  as  the  velocity 
is  greater.  Now  as  the  velocity  of  rotation  is  unlimited,  we  may 
always  conceive  it  to  be  so  great  that  the  centrifugal  force  shall  ex- 
ceed the  force  of  cohesion ;  in  which  case  the  particle  A  must  yield 
to  its  tangential  force  AF  and  be  detached  from  the  body.  When 
the  point  of  the  body  where  the  particle  is,  arrives  at  G,  the  particle 
itself  will  be  in  F;  and  consequently,  compared  with  the  place  where 
it  was,  will  be  farther,  reckoned  from  the  centre  C,  by  the  space 
GF;  which  is  a  manifest  effect  of  the  centrifugal  force. 

120.  The  effects  of  the  centrifugal  force,  combined  with  those  of 
a  central  force,  reciprocally  proportional  to  the  square  of  the  distance, 
completely  explain  the  slightly  oblate  form  of  the  planets,  and  the  di- 
minution of  gravity  at  their  equators.     The  principle  of  a  centrifugal 
force  enables  us  to  account  also  for  a  variety  of  phenomena  in  cur- 
vilinear motions,  which  present  themselves  every  day. 

(2.)  Oscillations  of  the  Pendulum. 

121.  When  a  heavy  body  is  attached  to  a  fixed  point,  about  which 
alone  it  can  turn,  it  cannot  remain  in  equilibrium,  unless  its  centre 
of  gravity  is  sustained;  that  is,  unless  it  is  in  the  vertical  plane  pass- 
ing through  the  axis  of  suspension. 


Oscillations  of  the  Pendulum.  39 

If  a  body,  placed  in  equilibrium  in  this  manner,  is  made  to  vibrate 
by  a  lateral  impulse,  its  motion  will  not  be  uniform.  It  may  be  put 
in  motion  by  gravity  alone,  without  an  impulse,  by  removing  it  more 
or  less  from  the  position  of  equilibrium,  so  that  the  centre  of  gravity 
be  not  in  the  vertical  line  passing  through  the  point  of  suspension.  If 
then  we  leave  the  body  to  itself,  oscillations  will  take  place ;  and 
this  species  of  motion  is  of  great  importance. 

122.  Every  body  AC  (jig.  12),  whatever  be  its  form,  is  called 
a  physical  or  compound  pendulum,  when  its  centre  of  gravity  B 
does  not  coincide  with  the  point  of  suspension  A.     A  simple  or  geo- 
metrical pendulum  is  a  single  straight  line  AB  (fig.  13),  turning 
about  A,   and  having  all  its  weight  concentrated  in  the  single  ex- 
treme point  B.     Strictly  speaking,  no  such  pendulum  exists  ;  but  a 
small  body  B  (fig.  14),  of  a  compact  mass,  suspended  by  a  6ne 
wire  JIB,  sufficiently  represents  it.    The  length  of  such  a  pendulum, 
considered  as  simple,  is  the  distance  from  the  point  of  suspension  A 
to  the  centre  of  gravity  B  of  the  body. 

123.  If  a  simple  pendulum  AB  (fig.  15)  be  drawn  into  the  posi- 
tion AC,  and  then  abandoned  to  itself,  the  material  point  B,  is  forced 
to  describe  the  arc  of  a  circle  CB.     It  describes  it  with  an  increas- 
ing velocity,  since  gravity  acts  continually  upon  it  at  each  point  of  its 
course  ;  but  as  the  direction  of  this  force  becomes  more  and  more 
oblique  to  its  motion,  the  acceleration  will  not  be  uniform,  but  will 
become  less  and  less  continually.     The  velocity  increases  from  C  to 
B,  where  it  is  at  its  maximum,  the  accelerating  force  becoming  noth- 
ing.    Accordingly,  the  body  will  not  remain  at  B,  but  will  continue 
to  describe  the  arc  BH,  by  virtue  of  its  force  of  inertia.     It  is  obvi- 
ous, however,  that  since  gravity  now  acts  in  a  contrary  direction,  its 
velocity  must  decrease  just  as  fast  as  it  before  increased  ;  so  that  it 
will  have  in  G,  for  example,  the  same  velocity  that  it  had  in  E,  at 
the  same  height.     If  we  draw  from  the  point  C,  where  the  motion 
commenced,  the  horizontal  line  CH,  it  is  evident  that  the  body  must 
rise  to  H;  but  at  H  it  will  be  in  the  same  state  as  at  C,  and  will, 
therefore,  return  from  A  to  C,  and  thus  continue  to  oscillate  between 
these  two  points.     Some  writers  understand  by  the  word  oscillation, 
a  single  passage  through  the  arc  of  vibration  ;  others  apply  it  to  two 
passages,  one  backward  the  other  forward.     We  shall  use  it  in  the 
former  sense. 

124.  It  is  evident  that  these  oscillations  would  continue  in  the 
same  manner  without  end,  if  there  were  no  obstacle  to  the  motion 


40  Solid  Bodies. 

but  the  resistance  of  the  air,  and  the  force,  however  inconsiderable, 
which  Is  necessary  to  bend  the  wire  at  A,  destroy  every  instant  some 
portion  of  the  velocity.  For  this  reason  the  length  of  the  arc  dimin- 
ished continually,  until  at  length  the  pendulum  ceases  to  vibrate. 
But  the  obstacles  to  the  motion  may  be  so  far  reduced,  that  these 
oscillations  will  continue  several  hours  in  succession.* 

125.  Though  it  is  very  easy  to  explain  the  manner  in  which  the 
motions  of  the  pendulum  are  produced,  it  is  very  difficult,  without 
the  aid  of  the  calculus,  to  present  an  exact  and  entire  theory  of  the 
instrument.  We  shall,  therefore,  content  ourselves  with  stating  the 
mathematical  results  deduced  from  experiment. 

(I.)  The  most  remarkable  property  of  this  species  of  motion  is 
the  perfect  equality,  or,  as  it  is  technically  called,  isochronism  of  the 
oscillations.  The  duration  of  an  oscillation  is  very  little  affected  by 
the  magnitude  of  the  arc  CB ;  and  as  the  arc  is  generally  one  of  a 
few  degrees,  in  experiments,  where  the  pendulum  is  employed,  the 
oscillations  are  to  our  own  senses  perfectly  isochronous,  f 

(2.)  In  equal  arcs  of  oscillation,  when  they  are  described  in  a 
vacuum,  the  weight,  the  magnitude,  the  form,  and  the  quantity  of 
matter  of  the  body,  have  no  effect  upon  th«  duration  of  an  oscillation. 

(3.)  The  time  of  an  oscillation  changes  with  the  length  of  the 
pendulum,  and  is  proportional  to  the  square  root  of  its  length.  J 

*  In  the  experiments  upon  the  length  of  the  pendulum,  made  at 
the  observatory  at  Paris,  with  the  apparatus  of  Borda,  the  motion 
was  perceptible,  by  means  of  a  microscope,  after  an  interval  of  24 
hours. 

t  The  time  of  an  oscillation  increases  somewhat  with  the  magni- 
tude of  the  arc  BC,  or  the  angle  BAG;  and  by  comparing  k  with 
that  of  an  infinitely  small  oscillation,  which  is  always  sensibly  the 
same,  the  increase  will  be  as  follows  ; 

If  CAB  =  30°  the  increase  will  be  0,01675 

15°  «  «         0,00426 

10°  «  «         0,00190 

5°  "  "         0,00012 

2*°  "  "         0,00003. 

The  time  of  an  infinitely  small  oscillation  is  here  taken  for  unity,  and 
in  an  arc  of  2£°  the  difference  is  extremely  small.  These  results 
are  obtained  from  the  formula  in  the  following  note. 

$  The  most  important,  but  at  the  same  time  the  most  difficult  prob- 
lem in  the  theory  of  the  pendulum,  is  that  which  relates  to  the  de- 


Oscillations  of  the  Pendulum.  41 

126.  It  may  be  inferred  from  these  laws  that  the  time  of  oscilla- 
tion of  the  simple  pendulum,  is  purely  the  effect  of  gravity,  independ- 
ently of  the  influence  of  every  other  force.  And  it  is  for  this  reason 
that  we  stated  in  Chapter  vu.,  that  the  pendulum  is  perfectly  fitted 
for  the  most  exact  researches  respecting  gravity. 

(4.)  From  this  consideration  we  deduce  the  fourth  law  ;  viz.  that 
the  time  of  oscillation  must  vary,  other  things  being  the  same,  when 


termination  of  the  time  em  loved  by  the  pendulum  in  making  any 
part  of  an  oscillation.  It  is  very  easy,  however,  to  determine  the 
velocity  of  a  body  at  each  point  ;  if  for  example,  the  bodv  passes 
from  C  to  E,  it  has  the  velocity  which  it  would  have  acquired  by 
falling  freely  from  D  to  F.  The  complete  determination  of  the  time 
cannot  be  obtained  except  by  a  complicated  integration,  of  whifh  we 
shall  give  simply  the  result.  Let  L  be  the  length  of  the  pendulum, 
T  the  time  of  an  oscillation  through  the  arc  CBH,  V  the  versed  sine 
of  the  angle  CAB,  g  the  force  of  gravity,  or  the  space  described  by 
a  heavy  body  in  the  first  second  of  its  fall,  and  n  the  semicircunifer- 
ence  of  a  circle  whose  radius  is  1,  or  3,1415926,  we  shall  have 


The  quantity  comprehended  within  the  parentheses,  forms  in  fact  an 
infinite  series  ;  but  it  converges  so  fast  that  in  very  small  arcs  the 
first  term  is  always  sufficient,  so  that  we  can  take 


IT 

71  \*g 


In  greater  arcs  we  have  occasion  to  employ  the  second  and  some- 
times even  the  third  term.  It  will  be  seen  that  g  and  L  will  be  both 
expressed  in  the  same  kind  of  measure,  which  is  the  unit  of  length, 
and  the  time  T  is  given  in  seconds,  since  g  answers  to  the  free  verti- 
cal descent  during  a  second  of  time. 

This  formula  serves  as  the  foundation  of  the  whole  theory  of  the 
pendulum.  We  shall  deduce  only  one  theorem  from  it.  If  we  raise 
to  the  square  the  approximate  value  of  T  and  multiply  it  hy  2g,  we 
obtain 

2gT*  =  n*  L. 

This  formula  may  he  employed  when  one  of  the  three  quantities 
T,  g,  and  L,  is  given,  to  find  the  third.  It  is  of  great  importance  in 
the  theory  of  gravity. — [See  Cam.  Mech.  art.  243,  et  seq.J 

Elem.  6 


42  Solid  Bodies. 

gravity  itself  varies.  The  time  will  be  longer  if  gravity  diminishes, 
and  shorter  if  it  increases.— [See  note  to  page  40.] 

127.  The  theory  of  the  compound  pendulum  presents  a  still 
greater  difficulty.  We  shall  only  speak  of  two  things,  viz.  the 
length  of  such  a  pendulum,  and  its  centre  of  oscillation. 

If  we  suspend  by  the  side  of  a  compound  pendulum  AC  (Jig.  12), 
a  simple  pendulum  JIB  (Jig.  14}  we  can  lengthen  or  shorten  them, 
so  as  to  cause  them  to  vibrate  in  equal  times.  If  then  we  take  the 
length  JIB  of  the  simple  pendulum,  and  apply  it  to  the  compound 
pendulum,  so  that  AD  (fg.  12)  =  JIB  (Jig.  14),  we  shall  find  that 
D  is  always  below  the  centre  of  gravity  B  of  the  compound  pendulum. 
There  are  even  cases  in  which  D  falls  entirely  without  the  body 
AC.  This  point  D  is  called  the  centre  of  oscillation  ;  and  AD  is 
called  the  length  of  the  compound  pendulum.  As  soon  as  we  have 
determined  with  sufficient  exactness,  the  centre  of  oscillation  of  a 
compound  pendulum,  it  may,  in  all  respects,  take  the  place  of  a  sim- 
ple pendulum. 


(3.)  Application  of  the  Pendulum. 

128.  The  pendulum,  on  account  of  the  isochronism  of  its  oscilla- 
tions, affords  the  best  means  of  measuring  time,  and  consequently  is 
best  adapted  to  the  purpose  of  regulating  clocks  and  watches.  Huy- 
gens  was  the  first  who  applied  it  in  this  way.  There  is  one  incon- 
venience attending  this  use  of  the  pendulum  arising  from  the  effect 
of  heat  and  cold  in  varying  its  length  ;  but  there  are  means  of  cor- 
recting; this  defect.* 


*  It  is  known  that  all  bodies  are  expanded  by  heat  and  contracted 
by  cold.  In  the  first  case,  the  lengthening  of  the  pendulum  lowers 
the  centre  of  oscillation,  and  the  oscillations  become  slower.  In  the 
second  case,  the  contraction  shortens  the  pendulum,  and,  the  centre 
of  oscillation  being  raised,  the  oscillations  are  more  rapid.  Means 
have  been  devised  to  make  this  cause  of  irregularity  correct  itself. 
[The  expansion  of  iron  and  brass  being  to  each  other  as  three  to  five, 
if  we  make  the  rod  FB  (Jig.  15)  of  iron,  and  the  rod  AO  of  brass  in 
the  proportion  of  5  to  3  ;  they  being  connected  at  the  lower  extrem- 
ities, and  the  weight  being  attached  at  O,  the  rod  AO  will  expand 
upward  just  as  much  as  the  rod  FB  expands  downward,  and  the 


Application  of  the  Pendulum.  43 

129.  The  application  of  the  pendulum  to  researches  respecting 
gravity,  is  still  more  important  for  the  progress  of  science.  Under 
this  head  we  remark  ; 

(1.)  That  the  invariable  isochronism  of  the  oscillations  in  the  same 
place,  demonstrates  the  constancy  of  gravity  itself. 

(2.)  It  is  proved  by  the  pendulum,  not  indeed  in  so  striking  a 
manner  as  by  experiments  in  a  vacuum,  but  with  still  greater  exact- 
ness, that  all  bodies  acquire,  by  gravity,  the  same  velocity  in  their 
fall ;  for  a  body  that  would  fall  more  slowly  than  another,  would,  if 
suspended  in  the  manner  of  a  pendulum,  perform  its  oscillations  with 
less  velocity. 

(3.)  The  simple  seconds  pendulum  furnishes  a  method  of  deter- 
mining the  space  described  by  a  heavy  body  in  the  first  second  of 
its  fall,  with  greater  precision  than  Atwood's  machine.  Indeed  it  is 
obvious  without  a  precise  demonstration,  that  a  determinate  ratio 
must  always  exist  between  the  length  of  the  simple  seconds  pendu- 
lum and  the  space  through  which  a  body  falls  in  a  second ;  since 
both  depend  simply  upon  gravity. 


point  O  where  the  weight  is  applied,  will  consequently  remain,  amid 
all  changes  of  temperature,  at  the  same  distance  from  F,  the  point  of 
suspension.  A  number  of  rods  of  each  kind  is  usually  employed,  as 
represented  in  figure  16,  where  the  rod  which  supports  the  weight, 
is  attached  at  F ,  and  free  at  D,  D',  the  brass  rods  expanding  upward 
and  the  iron  ones  downward,  as  before  ;  so  that  if  the  proper  propor- 
tion as  to  length  be  observed,  a  compensation  for  the  effect  of  tem- 
perature will  be  o  tained.  Other  means  have  been  invented  for 
accomplishing  the  same  purpose.  Of  these  we  shall  mention  only 
one  which  has  been  attended  with  great  success.  The  weight  AB 
(fig-  17r)  is  made  to  consist  of  a  glass  tube  about  two  inches  in  diam- 
eter, and  from  4  to  5  inches  long,  filled  with  mercury.  As  the  rod 
of  the  pendulum  supporting  the  weight,  expands  downward,  the  mer- 
cury expands  upward,  as  in  the  contrivance  first  mentioned,  and  the 
quantity  may  be  increased  or  diminished  till  a  compensation  is  effect- 
ed. A  clock,  provided  with  a  pendulum  of  this  construction,  made 
by  T.  Hardy  of  London,  for  the  Royal  Observatory  at  Greenwich, 
was  found,  after  two  years'  trial,  to  vary  only  }  of  a  second  in  24 
hours  from  its  mean  rate  of  going.  A  clock  of  the  same  construction, 
owned  by  W.  C.  Bond  of  Boston,  though-  much  less  costly,  has 
been  found,  by  careful  observation,  to  go  with  nearly  the  same  accu- 
racy.] 


Solid  Bodies. 


(4  )  The  opinion  of  Newton  respecting  the  diminution  of  gravity 
at  the  equator  is  perfectly  confirmed  by  obsevvations  on  the  pendu 
L.  A  pendulum  that  exactly  beats  seconds  here,  osc.llates  more 
Iwly  at  L  eqnator  and  more  rapidly  at  the  poles.  The  same  pen- 
d±,  therefore,  must  be  shortened  at  the  equator  and  .engthened 
at  the  poles,  in  order  to  beat  seconds  exactly.  Among  numerous 
observations  to  this  effect  we  select  the  following  ; 


[Place  of  Observation. 

St.  Thomas    . 

North  Latitude. 

.     .     0°24'  41" 

Length  of  the  Seconds  Fend. 

.     .     39,02074 

Trinidad    .     • 

.     .  10  38  56 

.     .     39,01884 

New  York      . 

.     .  40  42  48 

.     .     39,10168 

Hemmerfest   . 
Spitsbergen    . 

.     .  70  40  05 
.     .  79  49  58 

.     .     39,19519 
.     .     39,21469.] 

[It  is  evident  from  what  precedes,  that  the  value  of  g  answering 
to  different  latitudes,  may  be  exactly,  obtained  from  the  length  of  the 
pendulum  in  these  latitudes,  and  these  values  must  be  considered  as 
the  proper  measures  of  gravity  in  the  places  where  the  observations 
are  made.  The  difference  in  the  value  of  g  at  the  equator,  com- 
pared with  that  at  the  highest  latitude  yet  attained,  amounts  only  to 
about  one  inch.  It  may  be  easily  shown,  moreover,  that  the  length 
of  the  seconds  pendulum,  is  to  the  space  described  by  a  body  falling 
freely  in  one  second,  as  1  to  4,93480  ;*  so  that  one  of  these  results 
may  be  readily  deduced  from  the  other,  by  a  simple  proportion. 
Now  in  the  latitude  of  London,  at  the  level  of  the  sea  and  at  the 
temperature  of  60°  the  length  of  the  seconds  pendulum  is  found 
to  be  39,1386  inches  ;  hence  we  deduce  the  value  of  g  =  193,14 
inches,  or  16,1  feet,  nearly.] 

(5.)  On  very  high  mountains  the  oscillations  of  the  pendulum  are  a 
little  slower  man  on  the  general  level  of  the  earth's  surface.  Bou- 
guer  found  that  a  pendulum  which  made  98770  oscillations  in  24 
hours,  at  the  level  of  the  sea,  made  in  the  same  time  on  the  Pichin- 
cha,  only  98720.  Gravity,  therefore,  diminishes  as  we  remove  from 
the  surface  of  the  earth. 

(6.)  The  pendulum,  when  at  rest,  indicates  the  direction  of  gravi- 


*  [In  the  formula  2g  T2  =  n2  L,  of  a  preceding  note,  if  we  make 
T  equal  to  1,  we  shall  have  2  g  =.  n2  L,  which  gives  1  :  $  n2  : :  L  :  g, 
which  is  the  proportion  given  in  the  text.] 


Application  of  the  Pendulum.  45 

iy  in  the  most  exact  manner.  In  the  neighbourhood  of  vast  chains 
of  mountains,  it  has  been  found  that  its  direction  deviates  a  little 
from  the  vertical  towards  the  mountain  ;  this  is  a  manifest  proof 
of  the  existence  of  an  attractive  force  in  the  mountain,  which  is  ex- 
erted upon  the  body  of  the  pendulum.*  The  most  exact  observa- 
tions of  this  kind  were  made  in  Scotland  in  1774,  by  the  English 
astronomer,  Dr  Maskelyne.  He  calculated  the  attractive  force  of 
the  mountain  by  means  of  the  small  angle  through  which  the  plumb 
line  deviated  from  a  vertical  direction ;  and  compared  it  with  the 
attractive  force  of  the  earth,  as. ascertained  by  the  effect  of  gravity. 
This  enabled  him  to  compare  the  mass  of  the  mountain  with  the 
entire  mass  of  the  earth  ;  and  the  result  of  this  important  but  delicate 
inquiry  was,  that  the  mass  of  the  earth  is  about  4^  times  as  great  as 
the  mass  of  a  globe  of  water  of  the  same  bulk.  This  result  contra- 
dicts the  opinion  of  those  who  think  that  the  interior  of  the  earth  is 
filled  with  water.  Thus  it  is  to  the  pendulum  that  we  are  indebted 
for  some  important  inferences  respecting  the  nature,  or  at  least  the 
density  of  the  bodies  composing  the  interior  of  our  globe. f 

(7.)  Finally,  the  pendulum  may  be  applied  to  various  experiments 
upon  the  motions  of  bodies ;  because  we  can  easily  produce  by 
means  of  it,  motions  of  a  determinate  magnitude,  direction,  and  velo- 
city. It  may  be  demonstrated  by  geometry,  that  the  velocity  of  the 
pendulum  at  the  point  B  (jig.  18),  when  it  falls  from  different 
heights,  is  as  the  chord  of  the  arc  passed  through  ;  that  is,  the  ve- 
locity of  the  pendulum  in  5,  when  it  has  commenced  its  motion  in 
C,  is  to  the  velocity  in  B,  when  it  has  commenced  its  motion  in  E, 
as  the  arc  CB  is  to  the  arc  EB.  Now  it  is  easy  to  divide  the  arc 
BC  in  such  a  manner  that  the  chords,  reckoned  from  #,  shall  be  to 


*  This  deviation  is  proved  and  measured  by  observing  the  raeri- 
dion  distances  of  the  same  star  from  the  zenith,  on  both  sides  of  the 
mountain.  As  these  distances  are  reckoned  from  the  vertical,  deter- 
mined by  the  plumb  line,  we  can  ascertain  whether  they  are  the 
same  or  different.  For  the  attraction  of  the  mountain  tends  to  aug- 
ment the  one  and  diminish  the  other. 

t  Cavendish  arrived  at  the  same  result  by  a  very  simple  experi- 
ment. This  was  to  render  sensible  the  attraction  exerted  by  two 
large  globes  of  metal  upon  the  extremities  of  a  horizontal  lever,  sus- 
pended at  its  centre  by  a  wire  susceptible  of  torsion. — See  the  Me- 
caniqite  of  M.  Poisson,  vol.  ii.  p.  34. 


46  Solid  Bodies. 

each  other  as  the  numbers  1,  2,  3,  4,  and  5.  If  then  at  one  time 
we  raise  the  pendulum  to  12,  and  at  another  to  5,  the  velocities  in 
B  will  be  as  12  to  5,  &c.  An  arc  thus  divided  may  be  called  a 
scale  of  velocity.* 

In  these  circumstances  the  unit  of  velocity  is  not  determined,  and 
the  scale  only  gives  the  ratio  of  velocity ;  but  we  can  dispose  the 
apparatus  in  such  a  manner  that  the  scale  shall  indicate  in  inches  the 
absolute  velocity  acquired  by  the  pendulum.f 


*  Let  V,  V,  be  the  velocities  which  the  pendulum  acquires  by  its 
fall  through  the  arcs  CB  and  EB  ;  these  velocities  are  the  same  that 
a  body  acquires  by  a  free  descent  through  the  vertical  lines  DB  and 
FB  (108.)  We  have  then  V*  :  V*  ::  DB  :  FB  (100.)  Now  by 
a  known  property  of  the  circle  [Leg.  Geom.  art.  213,] 

2AB-.BC::  BC  :  BD  ;  also  2  AB  :  BE  :  :  BE  :  BF. 
Here  BC,  BE,  designate  the  chords  of  the  two  arcs.     Consequently, 


Hence  F2  :  V*  :  :  BC*  :  BE»  ;  or  V :  V  :  :  BC  :  BE ;  that  is,  the 
the  velocities  are  proportional  to  the  chords  of  the  arcs  through 
which  the  pendulum  falls. 

t  The  question  is  to  find  the  angle  BAC,  through  which  it  is  ne- 
cessary to  raise  the  pendulum,  in  order  that  it  shall  arrive  at  B  with 
a  velocity  V.  From  what  has  been  shown,  note  to  art.  101,  we  have 


. 

when  the  pendulum  is  to  have  in  B  the  velocity  V.    Now  the  radius 
of  the  described  arc  AB  =  L  ;  then 


But  in  the  triangle  CDA,  we  have  AC  :  AD  :  :  1  :  cos  BAC  ;  whence 


from  which  the  angle  BAC  may  be  obtained  by  the  trigonometri- 
cal tables.  If  in  this  formula  we  express  g  and  L  in  inches,  and  take 
successively  for  V,  the  natural  numbers  1,  2,  3,  4,  &c.,  we  shall  have 
awnes  of  angles  which  will  indicate  how  high  the  pendulum  must 
be  raised  in  order  to  have  in  C,  i,  2,  3,  4,  &c.,  inches  of  velocity. 


Communication  of  Motion  by  Impulse.  47 

CHAPTER  XV. 

Communication  of  Motion  by  Impulse. 

130.  SINCE  on  the  surface  of  the  earth  no  place  is  to  be  found 
absolutely  void  of  all  imp  netrable  jnatter,  every  body  in  motion  is 
continually  impinging  against  others.     Consequently,  we  cannot  ap- 
preciate any  motion  with  perfect  exactness,  if  we  are  unacquainted 
with  the  laws  according  to  which  bodies  communicate  their  motions 
to  each  other  by  impulse. 

131.  An  apparatus  is  made  use  of  to  ascertain  the  effects  of  im- 
pulse.    The  essential  parts  of  it  are  two  pendulums  AJ$  and  AC, 
(fig.  19.)  of  equal  length,  and  suspended  in  such  a  manner  that  the 
heavy  bodies  B  and  D  which  terminate  them  are  exactly  in  contact. 
Behind  them,  attached  to  the  same  support,  are  scales  of  velocity 
constructed   in  the   manner   prescribed  in  the  preceding  chapter. 
These  scales  may  be  traced  upon  the  arcs  described  from  the  points 
A  and  C  as  centres,  or  more  conveniently  upon  a  right  line  EF ; 
so  that  if  we  raise  one  of  the  pendulums  to  a  certain  point  G,  the 
number  at  G  will  indicate  the  velocity  which  the  pendulum  will  have 
at  the  lowest  point  of  its  arc,  where  it  will  impinge  against  the  mass 
D  of  the  other  pendulum.     We  may  thus  cause  one  pendulum  to 
impinge  against  the  other,  or  both  to  impinge  at  the  same  time,  and 
observe  the  phenomena  which  thence  result  in  their  motions. 


By  means,  therefore,  of  these  angles,  we  may  form  on  the  arc  BC,  a 
scale  of  absolute  velocities. 

The  isochronism  of  the  circular  pendulum  is  only  approximate ; 
it  is  true  only  for  very  small  arcs,  and  we  have  seen  that  the  os- 
cillations become  less  rapid  as  the  arcs  are  longer.  It  might  be 
proposed  to  find  what  kind  of  curve  would  make  the  isochronism 
exact  in  all  the  arcs.  Geometers  have  determined  this  to  be  a 
cycloid,  so  called,  because  it  may  be  generated  by  a  point  of  a  circle 
rolling  on  a  plane.  This  curve  is  such,  that  the  gravity  decomposed 
according  to  each  of  its  elements,  is  always  exacily  proportional  to 
the  arc  which  remains  to  be  described  before  reaching  the  lowest 
point ;  hence  arises  the  isochronism.  Attempts  have  been  made  to 
apply  this  curve  to  timekeepers  ;  but  they  have  been  abandoned 
on  account  of  the  difficulty  attending  its  rigorous  construction. 


48  Solid  Bodies. 

Another  appendage  consists  of  a  series  of  balls  arranged  one  after 
the  other,  the  object  of  which  is  to  show  in  what  manner  the  n 
of  one  or  several  is  transmitted  to  all  the  rest. 

A  third  consists  of  two  pendulums  so  arranged  as  to  be  capable  . 
impinging  at  the  same  time  against  a  third  body.     There  i 
another  appendage  to  percussion  machines,  by  which  heavy  bo 
are  made  to  fall  from  a  determinate  height  upon  a  hard  or  soft  body, 
for  the  purpose  of  observing  the  effects  of  such  collision. 

132.  According  to  Newton's  third  law  of  motion,  in  every  case  of 
the  collision  of  two  bodies,  a  transmission  of  motion  takes  place  from 
one  to  the  other  ;  but  we  cannot  determine  generally  the  quantity  of 
motion  transmitted,  because  this  depends  upon  a  great  variety  of  cir- 
cumstances, such  as  the  direction,  form,  mass,  velocity,  cohesive 
force,  elasticity,  state  of  aggregation,  &c.,  of  the  bodies  put  in  mo- 
tion.    Since  it  is  necessary  to  take  into  view  all  these  circumstances, 
the  theory  of  collision  must  be  of  great  extent,  and  in  some  particu- 
lars of  no  small  difficulty.     We  must,  therefore,  confine  ourselves  to 
the  most  remarkable  cases ;  and  chiefly  to  the  central  and  direct  im- 
pulses of  elastic  and  unelastic  bodies.     The  impulse  is  called  central 
when  the  bodies  move  before  collision  in  the  straight  line  passing 
through  their  centres  of  gravity,  and  when  the  collision  takes  place 
in  this  line.     It  is  called  direct,  when  the  surfaces  are  perpendicular 
to  the  direction  of  the  motion,  at  the  place  of  collision. 

133.  The  only  case  which  we  intend  to  examine  particularly,  is 
that  of  the  collision  of  unelastic  bodies,  because  this  is  the  basis  of 
the  whole  theory.     When  two  bodies  of  this  description  impinge 
against  one  another,  the  moving  body  communicates  to  the  body 
which  is  at  rest,  or  moving  in  an  opposite  direction,  the  quantity  of 
motion  necessary  to  give  both  an  equal  velocity.     The  same  is  true 
of  two  bodies,  of  which  one  moves  faster  than  the  other.    When  they 
have  acquired  the  same  velocity,  the  effect  of  collision  ceases,  be- 
cause no  pressure  can  take  place  between  them,  and  they  pursue 
their  course  together  and  with  an  equal  velocity,  as  if  they  formed 
one  body. 

The  case  which  most  frequently  occurs,  and  which  is  also  most 
easily  determined,  is  when  the  body  impinged  is  at  rest.  It  is  evi- 
dent, independently  of  calculation,  that  in  this  case  the  velocity  after 
collision  must  be  greatly  altered,  and  must  depend  principally  upon 
the  ratios  of  the  masses.  The  smaller  the  mass  of  the  body  imping- 
ed, compared  with  that  of  the  impinging  body,  the  less  will  be  the 


Communication  of  Motion  by  Impulse.  49 

force  necessary  to  put  it  in  motion,  and  the  less  will  the  velocity  of 
the  impinging  body  be  changed. 

On  the  contrary,  the  greater  this  mass  is,  the  less  will  be  the  velo- 
city after  collision.  If  the  mass  of  the  impinged  body  is  much  greater 
than  that  of  the  impinging  body,  the  motion,  after  collision,  will  never 
be  strictly  nothing,  but  may  be  absolutely  inappreciable  by  our 
senses,  as  when  a  hammer  is  struck  against  a  wall,  or  a  stone  is  suf- 
fered to  fall  upon  the  earth.* 

Among  the  circumstances  which  modify  the  effects  of  collision, 
we  shall  speak  first  of  the  force  of  cohesion.  The  impulse  exerts  an 
immediate  action  only  upon  the  parts  in  contact,  and  from  these 
it  is  propagated  to  the  other  parts.  This  is  done  more  rapidly 
in  proportion  as  the  bodies  are  more  hard  and  inflexible ;  and 


*  When  the  masses  and  velocities  of  the  bodies  before  collision  are 
known,  it  is  very  easy  to  find  by  calculation  the  velocity  after  col- 
lision. Let  m  be  the  mass  and  v  the  velocity  of  the  impinging  body, 
and.w'  the  mass  and  v'  the  velocity  of  the  body  impinged.  We  re- 
mark that  v'  is  to  be  taken  positively  when  the  bodies  move  in  the 
same  direction,  and  negatively  when  they  move  in  opposite  direc- 
tions. This  being  supposed,  the  sum  of  the  quantities  of  motion 
before  collision  will  be  mv  -f-  m'  v'.  After  collision,  the  two  bodies 
have  the  same  velocity  which  we  call  or  ;  and  the  mass  put  in  motion 
is  m  -J-  m'  ;  consequently,  the  quantity  of  motion  is  (m  -f-  m')  x. 
These  two  being  equal,  we  have 

/       i       /\  iii  m  v  -j-  m!  v' 

(m  -f-  m')  x  =  mv  -}-  m'  v',  or  x  =.  -  ~  —  -  —  . 

The  results  stated  in  the  text  are  deduced  rigorously  from  this  for- 

mula.    If  m1  is  at  rest  before  collision,  v'  =  0,  and  a;  =  --  -  —  .    If 

m-f-wt' 

v'  is  not  equal  to  zero,  but  m'  is  infinitely  small  compared  with  ///. 


that  is,  the  body  loses  only  an  infinitely  small  part  of  its  velocity.    If 
m,  on  the  contrary,  is  infinitely  small  compared  with  m', 

•     m1  v1 

•jTTr. 

that  is,  the  impinged  body  neither  gains  nor  losejs  -by  collision,  or  at 
least  loses  only  an  infinitely  •small  part  of  its  velocity. 
Elem.  7 


50  Solid  Bodies. 

more  slowly  in  proportion  as  their  particles  are  more  yielding,  for 
this  last  case  the  bodies  are  compressed  and  their  form  is  changed  j 
or  else,  if  the  force  of  the  cohesion  is  less  than  the  force  of  collis- 
ion, they  break  into  fragments  more  or  less  numerous.  Lastly,  the 
effects  are  variously  modified,  according  as  the  bodies  are  hard  or 
soft,  tenacious  or  friable,  &c. 

134.  Elasticity,  in  particular,  has  a  great  influence  upon  the  effect 
of  collision.  As  the  particles  of  elastic  bodies  yield,  the  bodies  are 
compressed  as  long  as  the  velocities  are  unequal.  Hence  the  effect  of 
collision  does  not  cease  in  this  case,  as  in  that  of  unelastic  bodies,  the 
moment  the  velocity  becomes  equal  in  the  two  ;  but  they  afterwards 
separate  from  one  another  because  they  are  forced  to  resume  their 
form ;  and  if  they  were  perfectly  elastic  they  would  do  it  with  ex- 
actly the  same  force  as  that  with  which  they  were  first  compressed. 

Thus  the  impinging  body  loses,  and  the  impinged  body  gains,  just 
double  the  motion  which  they  would  have  had,  if  they  had  been  un- 
elastic. Finally,  if  the  bodies  are  imperfectly  elastic,  the  loss  and 
augmentation  of  motion,  though  greater  than  for  unelastic  bodies,  are 
not  in  the  double  ratio,  as  is  the  case  with  perfectly  elastic  bodies.* 


*  Let  m  be  the  mass  of  the  impinging  body,  v  its  velocity  before 
collision,  and  u  its  velocity  after  ;  let  m1  be  the  mass  of  the  impinged 
body,  v'  its  velocity  before  collision,  and  u'  its  velocity  after.  If  the 
two  bodies  were  unelastic,  their  common  velocity  after  collision 
would  be  (133) 

_  m  v  -f-  m'  v' 


_ 

m 


and  m  would  have  lost  in  velocity  v  —  x.  This  loss  would  be  double 
in  bodies  perfectly  elastic,  that  is,  2  (v  —  z),  and  somewhat  larger 
than  v  —  a;  in  bodies  imperfectly  elastic.  Let  n  be  a  number  between  1 
and  2.  Then  the  loss  of  velocity  expressed  generally  will  be  n  (v  —  x)|; 
so  that  after  collision  there  remains  u  =.  v  —  n  (v  —  z).  In  like  man- 
ner, the  body  wi',  if  not  elastic,  will  gain  z  —  v',  by  collision  ;  if  per- 
fectly elastic,  2  (x  —  v'}  ;  and  generally  n  (z  —  »').  Its  velocity  after 
collision  will,  therefore,  be  u'  =  v'  +  n'  (x  —  v1).  If  in  the  values 
of  u  and  u'  we  substitute  for  z  its  value,  found  above,  we  shall  have, 
by  a  simple  transformation, 


These  two  formulas  are  of  very  general  use  ;  if  we  suppose  n  =  2, 


Communication  of  Motion  by  Impulse.  §1 

135.  The  variations  depending  upon  the  direction  of  the  impulse 
are  still  more  numerous.     We  shall  only  state  this  single  law,  that  at 
each  eccentric  impulse  there  is  always  produced  a  circular  motion 
about  the  centre  of  gravity,  which,  in  most  cases,  renders  the  mathe- 
matical estimation  of  the  effect  very  difficult.     The  law,  at  the  same 
time,  is  so  general,  that  when  two  bodies  are  connected  together  by 
a  visible  line,  or  even  by  an  invisible  attractive  force,  no  partial  mo- 
tion of  one  body  is  possible,  without  being  felt  by  the  other  ;  and  if 
one  of  them  is  put  in  motion,  both  begin  to  turn  about  their  common 
centre  of  gravity.     Thus  the  moon  and  earth  move  about  their  com- 
mon centre  of  gravity.     In  like  manner  the  planets  do  not,  strictly 
speaking,  move  simply  round  the  sun,  but  both  the  sun  and  planets 
move  together  about  the  centre  of  gravity  of  the  solar  system. 

136.  We  shall  here  state  only  a  single  case  of  oblique  collision. 
If  an  elastic  ball  A,  (fig.  20)  is  thrown  in  a  direction  BA  against  an 
elastic  wall,  experience  teaches  that  it  rebounds  in  a  direction  AH, 
under  an  equal  Bangle.     To  explain  this  effect  let  BA  represent  the 
force  of  collision.  Decompose  this  into  two  others,  one  FA,  parallel  to 
the  wall  CD,  and  the  other  EA  perpendicular  to  it.  The  only  effect 
of  EA)  if  it  acted  alone,  would  be  to  make  the  body  rebound  with 
an  equal  force  AE,  in  the  direction  AE.     But  the  other  force  FA, 
having  no  obstacle,  would,  if  it  acted  alone,  impel  the  body  in  the 
direction  AG.     Now  since  these  two  act  at  the  same  time  upon  the 
body,  it  must  describe  the  diagonal  AH.     In  general,  the  oblique 
and  the  eccentric  impulse  reduce  themselves  by  means  of  the  de- 
composition of  forces  to  the  laws  of  central  and  direct  impulses. 

137.  We  have  also  mentioned  the  state  of  aggregation  of  bodies, 
among  the  circumstances  which  modify  the  effect  of  collision.     If  a 
solid  body  moves  in  a  fluid,  or  if  one  fluid  moves  in  another,  these 
bodies  are  in  constant  collision  with  respect  to  each  other.     But  as 
this  difficult  branch  of  the  theory  of  collision  belongs  to  the  examina- 
tion of  liquid  and  aeriform  bodies,  we  shall  only  remark  here,  that 
the  equality  of  action  and  reaction  which  constitutes  Newton's  third 
law  of  motion,  is  still  observed  in  these  motions ;  for  in  conformity 
with  this  principle,  the  body  put  in  motion  loses  just  as  much  motion 

they  are  applicable  to  perfectly  elastic  bodies.  If  we  put  n  =  1, 
they  answer  for  unelastic  bodies.  Finally,  if  the  bodies  have  an  im- 
perfect elasticity,  n  has  an  intermediate  value  which  may  be  ascer- 
tained by  experiment. 


62  Solid  Bodies. 

as  it  communicates  to  the  fluid  medium.     Regard  must  be  paid  to 
this  law,  if  we  would  judge  properly  of  any  motion  which  takes  pla< 
in  the  air  or  in  the  water.* 


CHAPTER  XVI. 

Vibratory  Motions,  and  the  Sounds  which  they  produce  ;  or  the  first 
Principles  of  Acoustics. 

138.  ALL  we  know  respecting  the  production  of  sound  is  derived 
from  the  observation  of  solid  sonorous  bodies.     This  is  the  reason 
why  we  treat  of  acoustics  in  this  place,  and  not  as  is  generally  done, 
under  the  head  of  air. 

139.  If  an  elastic  wire  chord  stretched  between  AB  (fig.  21)  be 
drawn  out  of  its  rectilinear  direction  and  then  left  to  itself,  it  does  not 
on  its  return  remain  in  this  direction,  but  like  a  pendulum  moved  out 
of  the  vertical  line,  it  passes  beyond  its  natural  position  at  rest,  then 
back  again,  taking  first  the  position  ADB,  then  ACB  on  the  oppo- 
site side.     This  motion,  performed  with  great  rapidity,  is  called  an 
oscillation  or  vibration.     The  theory  of  these  motions  is  a  difficult 
one.     For  this  reason  we  shall  give  a  simple  statement  of  the  laws 
confirming  them  by  experiment.! 

140.  The  vibrations  of  a  chord  have  this  in  common  with  the 
oscillations  of  a  pendulum,  thai  they  are  almost  exactly  isochronous. 
The  time  of  a  vibration  depends  entirely  upon  the  length,  weight, 
and  tension  of  the  chord  .J 

141.  When  the  vibrations  are  very  quick,  we  hear  a  determinate 


*  [For  a  more  complete  analysis  of  the  laws  of  collision,  see  Cam. 
Mech.  art.  292,  &c.] 

t  Some  writers  apply  the  term  vibration  to  the  internal  motion  of 
the  smallest  particles  of  a  body,  by  which  they  actually  change  their 
respective  situations,  though  in  a  manner  imperceptible  to  our  senses. 
Undoubtedly  there  are  such  motions,  but  they  never  produce  any 
sound. 

|  Let  T  be  the  time  of  a  vibration ;  g  the  force  of  gravity  ;  7  the 
length  of  the  chord ;  w  its  weight ;  and  /  the  force  with  which  it  is 
stretched.  We  shall  have 


Vibratory  Motions. 


sound,  which  is  more  or  less  grave  or  acute,  according  to  the  velo- 
city of  the  vibrations. 

142.  There  is  a  very  convenient  instrument  for  illustrating  the 
laws  of  these  vibrations  called  a  monochord.      It  consists  of  one 
or  a  small  number  of  chords  stretched  over  a  wooden  table  in  such 
a  manner  that  their  length  and  tension  may  be  varied  at  pleasure. 
In  making  experiments  with  this  instrument,  the  following  principle, 
capable  of  being  demonstrated  by  the  calculus,  must  be  admitted. 
The  times  of  oscillation  of  the  same  chord,  other  things  being  the 
same,  are  proportional  to  its  length. 

143.  By  means  of  the  monochord  it  is  shown  that  there  is  a  par- 
ticular musical  interval  answering  to  each  ratio  in  the  oscillations  or 
lengths  of  the  chord.     These  are  denominated  as  follows  ; 


'   1 

2 

octave, 

2 

3 

fifth, 

3 

4 

fourth, 

4 

5 

major  third, 

5 

6 

minor  third, 

6 

7 

superfluous  second, 

7 

8 

second, 

For  the  ratio  .  .  .    < 

8 

9 

9 

10 

entire  tone, 

10 

11 

11 

12 

12 

13 

13 

14 

semitone. 

14 

15 

15 

16 

-J 


**/ 

This  formula  serves  generally  for  all  kinds  of  chords. 

Let  us  suppose  a  cylindrical  chord  of  the  same  thickness  and  elas- 
ticity through  its  whole  length.  Let  r  be  half  the  diameter,  and  d  the 
specific  gravity  of  the  substance.  If  we  call  n  the  semicircumference 
of  a  circle  whose  radius  is  1,  the  bulk  of  the  chord  will  be  nr2  lt 
and  its  weight  w  ~  n  ra  IS.  If  we  substitute  this  value  in  the  for- 
mula above  given,  we  shall  have 

"TF 


,=     \nr2l* 
N  2*/ 


This  formula  is  as  general  as  the  preceding,  and  is  more  convenient 
for  chords  which  are  homogeneous  throughout. 


54  Solid  Bodies. 

The  ratios  expressed  by  more  compound  numbers  are  of  little  use 
in  music.  The  5  first,  together  with  the  ratio  of  3  :  5  which  gives 
the  sixth,  are  called  concords  or  consonant  intervals.  The  others  are 
called  discords. 

144.  A  chord  renders  the  tone  which  is  proper  to  it,  either  by 
striking  it  or  passing  over  it  the  bow  of  a  violin.     In  the  first  case,  a 
practised  ear  perceives,  besides  the  fundamental  tone  of  the  chord, 
a  variety  of  relative  tones.     If  we  call  1  the  time  of  vibration  of  the 
fundamental  tone,  |  will  be  the  time  of  vibration  of  the  nearest  rela- 
tive tone,  and  the  following  will  be  |,  j,  j,  j,  j,  &c.     These  rela- 
tive tones  are  thus  accounted  for.     While  the  entire  chord  performs 
one  oscillation  (Jig.  21),  its  half  (fig.  22),  its  third  (fig.  23),  its 
fourth  (jig.  24),  vibrate  also,  which  renders  the  compound  motion 
of  the  chord  veiy  complicated.     When  we  pass  a  bow  over  the 
chord  it  is  not  apparent  that  such  relative  tones  are  produced.     The 
points  G  (Jig.  22)  ;  K,  L,  (Jig.  23) ;  O,P,Q,  (fig.  24)  ;  in  which 
the  direction  of  the  vibration  changes,  are  called  nodes  of  oscillation. 
We  can,  by  a  certain  address,  produce  only  relative  tones. 

145.  It  is  possible,  by  means  of  a  bow,  to  obtain  tones  not  only 
from  chords  but  from  ah1  bodies  however  little  elastic.     Chladni,  to 
whom  we  are  indebted  for  so  many  valuable  discoveries  in  acoustics, 
has  given  a  method  of  rendering  almost  visible  the  oscillations  of 
plates  and  many  other  bodies,  by  sprinkling  sand  of  equal  thickness 
over  them. 

146.  Next  to  the  monochord,  nothing  is  more  convenient  for  ex- 
periments upon  sound,  than  the  vibrations  of  elastic  bars,  fixed  at 
one  of  their  extremities.     They  may  be  taken  long  enough  to  enable 
us  to  count  the  vibrations.     By  this  means  the  following  law  may  be 
demonstrated ;  that  the  times  of  oscillation  decrease  as  the  squares  of 
the  lengths.     By  diminishing  the  thickness,  we  are  enabled  to  pro- 
duce oscillations  so  rapid  as  to  give  a  sound  sensible  to  the  ear.   We 
may  consequently  determine,  by  means  of  the  calculus,  the  times  of 
oscillation  which  escape  observation.      Finally,  by  taking  this  in 
connexion  with  the  monochord,  we  may  confirm  all  the  results  of  the 
theory  of  sound. 

147.  It  is  demonstrated,  by  experiments  of  this  kind,  that  the 
lowest  appreciable  tone  is  that  which  is  produced  by  a  chord  per- 
forming about  32  vibrations  in  a  second.     Each  of  the  octaves  to 
this  sound  corresponds  to  a  number  of  vibrations,  double  that  which 
precedes  it ;  whence  we  obtain  the  following  series  ; 


Vibratory  Motions.  55 

Lowest  tone  32  oscillations  in  a  second 

1st  octave  64  " 

2d       "  128  "  " 

3d       "  256  "  " 

4th       "  512  "  " 

5th      "  1024  "  " 

6th      "  2048  " 

There  is  also  as  to  acuteness  a  limit  t^> appreciable  tones.  The 
9th  octave  above  the  lowest  tone,  is  considered  as  the  highest  tone 
which  can  be  sensible  to  the  ear. 

148.  Although  this  is  not  the  place  to  speak  of  the  properties  of 
the  air,  we  must  remark  that  this  fluid  possesses  a  high  degree  of 
elasticity ;  that  is,  it  is  capable  of  great  compression,  and  when  the 
pressure  ceases,  of  a  correspondent  dilatation.     It  will,  therefore,  be 
easily  seen,  that  the  vibrations  of  a  sonorous  body  must  necessarily 
produce  a  similar  vibratory  motion  in  the  air,  which  will  propagate 
itself  to  a  great  distance  from  the  sonorous  body.     Thus  when  a 
chord  JIB  (fg.  25)  begins  to  vibrate,  it  is  obvious  that  there  will  be 
produced  about  the  chord  strata  of  compressed  air  Jl  c  B,  Ad  B, 
A  e  B,  JlfB,  &c.,  which  will  alternate  with  the  strata  of  rarefied 
air.  These  alternate  compressions  and  rarefactions,  which  are  called 
pulses  or  undulations,  succeed  each  other  with  great  rapidity  near 
the  chord,  without  causing  the  separate  particles  of  air  which  com- 
pose them  sensibly  to  change  their  place.     This  motion  has  a  strik- 
ing resemblance  to  the  circular  undulations  produced  by  throwing  a 
stone  into  a  tranquil  body  of  water.     The  motion  of  the  air  being 
propagated  to  the  ear,  we  have  the  sensation  of  sound. 

149.  Experience  teaches  us  that  all  sounds  are  propagated  widi 
the  same  velocity,  whatever  be  the  rapidity  of  the  vibrations.     In 
atmospheric  air,  this  velocity,  according  to  the  most  exact  computa- 
tions, is  1142  feet  in  a  second  ;  but  sound  propagates  itself  also 
through  water  and  solid  bodies,  and  with  a  still  greater  velocity,  as 
is  proved  by  experiment. 

150.  As  to  the  force  of  sound,  it  is  proved  by  observation,  that 
it  diminishes  as  the  air  is  more  rarefied,  and  increases  as  the  air  be- 
comes denser.     It  decreases  also  with  the  distance,  and  probably  in 
the  inverse  ratio  of  the  square  of  the  distance. 

151.  It  is  difficult  to  determine  by  experiment  whether  sound  is 
propagated  in  a  straight  liue  only,  or  whether  it  is  also  propagated  in 


56  Solid  Bodies. 

curvilinear  directions,  since  the  judgments  which  our  senses  form, 
respecting  the  direction  of  sound,  are  attended  with  great  uncer- 
tainty. The  reflection  of  sound  produced  by  solid  bodies,  according 
to  the  laws  of  collision  of  elastic  bodies,  renders  the  first  opinion 
probable.  It  is  to  this  reflection  or  reverberation  that  we  are  to  as- 
cribe the  phenomena  of  echos,  the  speaking  trumpet,  whispering 
galleries,  &tc. 

152.  In  wind  instruments,  it  is  not  the  solid  substance   of  the 
instrument,  but  the  coluien  of  confined  air  which  forms  the  sono- 
rous body.     But  the  particular  properties  of  these  vibrations  are  not 
yet  sufficiently  explained. 

153.  There  are  many  modifications  of  sound  besides  what  we 
call  high  and  low,  loud  and  soft,  concerning  which  little  can  be  said 
with  certainty,  although  the  ear  distinguishes  them  with  great  accura- 
cy.   Among  these  modifications  may  be  mentioned  the  peculiarity  in 
the  sound  of  different  instruments,  and  of  different  voices.    Articulate 
sounds,  uttered  by  the  human  organs,  are  particularly  remarkable. 
They  are  called  vowels  and  consonants. 


SECTION  III. 

HEAT. 


CHAPTER  XVII. 

Heat  in  general ;  its  expansive  Force  ;  Thermometer  and  Pyrometer. 

1 54.  HEAT,  the  presence  of  which  is  always  known  by  a  particu- 
lar sensation,  and  the  principal  gradations  of  which  are  expressed  by 
the  words,  heat,  warmth,  cold,  acts  a  very  important  part  in  nature. 
By  its  diminution  almost  all  liquid  substances,  and  even  many  aeri- 
form bodies,  become  solid.     By  its  increase  almost  all  liquid  bodies 
and  many  solids  become  aeriform.     Without  heat  there  would  be 
no  life  or  organization.     Indeed  the  use  of  heat  in  providing  for 
our  natural  and  artificial  wants,  is  so  various  and  important,  that  if  it 
were  withdrawn,  we  should  return  to  a  level  with  the  brute  creation. 
For  these  reasons  it  may  well  be  regarded  as  one  of  the  most  im- 
portant subjects  of  philosophical  inquiry. 

155.  The  cause  of  heat  is  beyond  the  reach  of  our  senrcs.    Some 
philosophers  are  inclined  to  ascribe  it  to  an  internal  motion  of  the 
smallest  particles  of  bodies.     Chemists  are  unanimous  in  ascribing  it 
to  the  existence  of  a  peculiar  substance,  which  they  call  caloric.   We 
shall  find  hereafter,  if  not  decisive  proofs,  at  least  strong  reasons  in 
support  of  this  opinion.    In  the  mean  time  we  shall  employ  the  word 
caloric  as  a  convenient  mode  of  expression. 

156.  The  first  effect  of  heat  which  we  shall  notice,  is  that  it 
causes  all  bodies  to  expand  ;  solids  little,*  liquids  more,f  and  aeri- 
form bodies  most  of  all.  J 

*  The  expansion  of  solid  bodies  may  be  proved  by  a  very  simple 
experiment.  Let  there  be  a  solid  body  AB,  (Jig.  26)  which,  when 
cold,  exactly  passes  between  the  two  vertical  columns  CD,  EF.  If 
we  heat  this  bar  without  heating  the  apparatus  CDEF,  it  will  no  lon- 
ger be  contained  between  the  vertical  columns,  but  will  take  the  ob- 
lique position  represented  in  the  figure. 

t  If  we  fill  a  vessel  AB,  (Jig.  27)  with  a  liquid,  and  insert  a  tube 
CD,  open  at  both  ends,  taking  care  to  close  entirely  the  orifice  F, 
upon  the  application  of  heat,  the  water  will  rise  and  fill  the  tube. 

J  If  we  only  put  a  sufficient  quantity  of  water  in  the  vessel  to 

Elem.  8 


58  Heat. 

157.  This  circumstance  suggests  a  very  natural  and  simple  method 
of  measuring  with  great  precision  the  increase  and  diminution  of 
heat.     Instruments  used  for  this  purpose  are  called   thermometers. 
The  first  person  who  contrived  an  instrument  of  this  kind  was  Dreb- 
bel  of  Holland,  but  it  was  very  imperfect.     This  was  toward  the 
end  of  the  16th  century.     In  the   17th  century  the  academicians 
of  Florence  improved  its  construction.     Finally,  in  the  18th,  Fah- 
renheit, at  Dantzick,  and  Reaumur  in  France,  discovered  at  the 
same  time  the  exact  principles  upon  which  these  instruments  are  con- 
structed. 

158.  The  apparatus  most  in  use  at  present,  is  that  which  is  with 
good  reason  called  Delve's  thermometer,  his  ingenious  researches 
having  greatly  contributed  to  its  perfection.     The  following  is  a  de- 
scription of  its  essential  parts.     The  lower  end  of  a  hollow  glass 
tube  (fig.  28),  is  blown  into  a  round   bulb.    This  bulb   is  then 
heated,  in  order  to  dilate  the  enclosed  air,  the  upper  orifice  being 
left  open.     This  being  done,  the  tube  is  inverted  and  the  open  end 
immersed  in  mercury.     As  the  enclosed  air  cools  and  condenses, 
the  mercury  is  forced  .into  the  tube  by  the  hydrostatic  pressure  of  the 
external  air,  as  we  shall  show  hereafter.     When  the  tube  and  a  part 
of  the  bulb  are  filled  with  mercury,  the  position  is  reversed,  and  the 
bulb  is  plunged  into  boiling  water.     The  mercury  immediately  rises 
to  a  certain  fixed  point  E,  called  the  boiling  point,  and  remains 
there  as  long  as  the  bulb  continues  in  boiling  water.     In  the  next 
place  the  bulb  is  surrounded  with  melting  ice.     The  mercury  now 
descends  to  a  point  G,  where  it  remains  invariably  fixed,  as  long  as 
the  bulb  is  surrounded  with  ice.     This  is  called  the  freezing  point. 
The  distance  between  the  two  points  thus  determined,  is  called  the 
fundamental  interval.     The  tube  is  then  attached  to  a  small  frame 
or  scale,  on  which  the  divisions  are  marked.     The  fundamental  in- 
terval is  divided  into  80  parts  in  Reaumur's  thermometer ;  and  the 
divisions  are  continued  above  E  and  below  G,  as  far  as  the  tube  ex- 


cover  the  lower  opening  of  the  tube,  and  heat  the  enclosed  air,  the 
water  will  rise  very  sensibly  in  the  tube.  A  still  better  way  is  to 
leave  nothing  but  air  in  the  vessel,  and  after  having  closed  with  great 
care  the  orifice  between  the  lube  and  the  mouth  of  the  vessel,  to 
insert  into  the  tube  a  drop  of  coloured  liquid.  The  least  cooling  of 
the  enclosed  air  will  cause  the  drop  to  descend  and  p-ecipitate  itself 
on  the  bottom  of  the  vessel.  On  the  contrary,  the  least  additional 
heat  will  cause  it  to  ascend  and  escape  from  the  tube. 


Thermometer.  59 

tends.     The  point  G  is  marked  zero,  and  from  this  point  we  count 
in  both  directions. 

159.  If  we  divide  GE  into  180  parts  and  place  zero  at  the  point 
K,  32  divisions  below  Cr,  and  then  count  each  way  from  this  point, 
the  point  G  will  be  marked  32,  and  E  212.  This  is  the  thermom- 
eter of  Fahrenheit.  Reaumur's  thermometer  is  usually  filled  with 
alcohol  instead  of  mercury.  The  new  French  scale  is  the  same  as 
the  Swedish  scale  of  Celsius,  in  which  the  fundamental  interval  is 
divided  into  a  hundred  parts,  zero  being  placed  at  the  freezing 
point.* 

*  This  method  supposes  that  the  bore  of  the  tube  is  exactly  cylin- 
drical, so  that  equal  quantities  of  mercury  correspond  to  equal  divis- 
ions. But  in  reality  this  is  never  the  case,  whatever  care  be  taken 
in  the  choice  of  the  tubes.  And  hence  there  are  few,  if  any,  per- 
fectly accurate  thermometers.  Gay-Lussac,  who  has  made  many  val- 
uable experiments  on  the  dilatation  of  gases,  had  occasion  for  the 
most  perfect  thermometers,  and  the  method  he  used  was  the  fol- 
lowing. 

Take  a  glass  tube  open  at  both  ends,  and  introduce  a  little  mercu- 
ry ;  this  will  form  a  small  column  in  the  interior  of  the  tube.  Mark 
on  the  glass  the  extreme  points  where  the  column  ends,  and  apply 
this  distance  successively  throughout  the  whole  length  of  the  tube, 
beginning  at  one  extremity.  We  shall  thus  have  a  first  scale  of  equal 
parts,  the  length  of  one  of  which  we  call  /. 

Then  take  out  a  portion  of  the  mercury  before  introduced  ;  for 
instance,  a  little  less  than  half.  The  remaining  column,  placed  at 
one  end  of  the  tube  will  not  fill  up  one  of  the  first  divisions  /,  but 
will  exceed  half  of  it.  Mark  the  point  where  it  terminates.  Let 
the  column  now  be  placed  so  that  one  of  its  extremities  shall  coin- 
cide with  the  mark  of  the  first  division.  The  other  will  not  reach 
the  mouth  of  the  tube,  but  as  before,  will  take  up  more  than  half  of /; 
mark  the  point  where  it  terminates.  This  point  and  the  preceding 
will  be  equally  distant  from  the  middle  of  the  interval  /.  Accordingly, 
we  may  obtain  the  middle  by  bis.  cting  the  space  which  separates 
them  ;  for,  if  we  have  taken  out  nearly  half  the  mercury  first  intro- 
duced, and  if  the  tube  is  not  very  unequal,  the  distance  between  the 
two  points  must  be  very  small,  and  for  so  small  an  extent  the  tube 
may  safely  be  considered  as  cylindrical.  We  repeat  this'  process  for 
each  of  the  first  divisions,  and  thus  obtain  a  second  scale,  containing 
twice  as  many  intervals  as  the  first.  Each  of  these  second  intervals 


6tf  Heat. 

160.  The  air  thermometer  consists  of  a  tube  ABC,  (fig.  29) 
recurved  at  B,  and  terminated  by  a  bulb  C.  The  bulb  .partly 
filled  with  air  5  the  rest  of  the  space  contains  mercury  winch 
nearly  to  the  middle  of  the  long  branch  of  the  tube.  When  the  air 
is  heated  at  C,  it  expands  and  the  mercury  rises;  when  cooled,  it 
contracts,  and  the  mercury  descends.  If  the  points  of  freezing  and 
boiling  are  determined  as  above,  and  the  fundamental  interval  divid- 
ed into  370  parts,  we  shall  have  the  thermometer  of  Lambert. 

is  to  be  again  divided  as  before  into  two,  and  these  again  into  two 
others,  and  so  on  until  we  have  the  number  of  divisions  requ.red. 

This  being  done,  let  a  bulb  be  formed  at  the  end  of  the  tube,  and 
introduce  the  mercury  as  pure  and  dry  as  it  can  be  obtained.  The 
mercury  is  to  be  boiled  in  the  tube  itself;  after  which  the  thermom- 
eter is  completed  in  the  ordinary  way,  by  marking  the  points  at  which 
ice  begins  to  melt  and  water  to  boil.  The  number  of  divisions  found 
between  these  points  will  indicate  the  scale  of  the  thermometer; 
which  being  constructed  upon  these  principles  will  be  perfectly  ac- 


curate. 

It  is  easy  to  reduce  these  degrees  to  those  of  Reaumur  or  Fahren- 
heit, or  any  other  proposed  scale  ;  for  let  n  be  the  number  of  divis- 
ions of  the  fundamental  interval ;  it  is  obvious  that  each  degree  of 

this  scale  will  be  equal  to  —  of  Reaumur,  —  of  Fahrenheit,  and 

of  the  centesimal  scale. 

n 

In  fixing  the  boiling  point,  care  should  be  taken  at  the  same  time 
to  note  the  height  of  the  barometer,  which  measures  the  weight  of 
the  atmosphere.  For  when  the  barometer  is  low,  water  boils  at  a 
less  heat  than  when  it  is  high,  as  will  be  seen  in  the  following  chap- 
ter. We  have  been  thus  particular  respecting  the  thermometer,  be- 
cause it  is  an  instrument  of  such  extensive  use. 

*  When  we  wish  to  measure  with  great  exactness  very  small 
changes  of  temperature,  we  employ  another  sort  of  air  thermometer. 
We  take,  as  before,  a  glass  tube  terminated  by  a  hollow  bulb  ;  but 
instead  of  introducing  mercury,  we  make  use  of  the  air  which  the 
tube  contains,  and  measure  variations  of  temperature  by  the  changes 
of  its  bulk.  For  this  purpose  we  separate  this  portion  of  air  from 
the  air  without,  which  is  done  by  heating  the  air  in  the  bulb, 
holding  it  in  the  hand.  The  warmth  of  the  hand  expands  the  inter- 
nal air  and  expels  a  portion  from  the  tube.  Then  we  put  a  small 


Thermometer.  61 

161.  In  constructing  the  thermometer,  mercury  has  the  following 
advantages.  1.  It  supports,  before  boiling,  more  heat  than  any 
oilier  fluid ;  and  by  employing  it  we  may  extend  the  scale  beyond 
the  point  of  boiling  water  to  252  of  Deluc,  and  600  of  Fahrenheit. 
Below  the  above  point  it  may  be  extended  to  32  of  Deluc,  and 
—  40  of  Fahrenheit.  At  this  point  the  mercury  becomes  solid.  2. 
Mercury  may  be  obtained  perfectly  pure  and  similar  in  its  properties 
more  easily  than  any  other  fluid  ;  and  hence  the  results  given  by 
different  thermometers  may  be  more  safely  compared  with  each 
other.  3.  Mercury  is  more  sensible  to  the  action  of  heat  than  any 
other  fluid,  that  is,  it  indicates  more  promptly  the  effects  of  heat  and 
cold.  4.  But  its  essential  superiority  arises  from  the  fact,  that  its 
dilatation  is  nearly  proportional  to  the  actual  increase  of  heat  ;  at 
least  between  the  freezing  and  boiling  points.  Deluc  demonstrated 
this  fact  by  a  series  of  very  exact  experiments.* 

drop  of  coloured  spirits  of  wine  into  the  orifice  of  the  tnbe,  arid  suf- 
fer the  air  within  to  cool  and  contract ;  in  consequence  of  which  the 
drop  descends  into  the  tube  ;  and  afterwards  rises  or  falls  with  the 
smallest  change  of  temperature.  The  sensibility  of  this  apparatus 
depends  upon  the  ratio  of  the  tnbe  and  ball,  and  may  be  estimated 
according  to  the  known  laws  of  dilatation.  But,  on  account  of  this 
extreme  sensibility,  it  can  only  be  employed  within  very  narrow  lim- 
its ;  for  if  the  cooling  is  very  considerable,  the  drop  falls  into  the  ball ; 
and  if  it  is  much  heated  it  is  expelled  from  the  tube. 

If  two  balls  be  thus  formed,  one  at  each  extremity,  and  a  coloured 
drop  be  introduced  through  a  small  perforation  afterwards  closed, 
the  drop  will  be  influenced  only  by  the  difference  of  temperature  of 
the  two  masses  which  it  separates.  This  instrument,  which  is  used  in 
many  delicate  experiments,  is  called  the  Thermoscope  or  Differential 
Thermometer. 

*  When  we  mix  two  portions  of  water  of  equal  weight,  but  of  differ- 
ent temperatures,  a  thermometer  plunged  into  the  mixture,  ought  to 
indicate  the  degree  exactly  intermediate  between  the  two  degrees  of 
heat,  provided  its  expansion  is  exactly  proportional  to  the  degree  of 
heat.  On  this  fact  is  founded  the  method  employed  by  Deluc,  in 
comparing  the  range  of  a  mercurial  thermometer  with  that  of  heat. 
This  experiment  is  very  difficult,  because  in  order  to  be  accurate,  it 
is  necessary  that  the  bodies  operated  upon,  should  be  completely  in- 
sulated from  all  foreign  bodies.  But  it  being  a  truth  of  great  import- 


Heat. 


Pyrometer. 

162.  Various  instruments  have  been  invented  for  measuring  very 
high  degrees  of  heat.  These  are  called  pyrometers.  Most  of  them 
are  founded  upon  the  dilatation  of  solid  bodies,  and  principally  of 

ance,  many  efforts  have  been  made  to  establish  it.     Gay-Lussac  at 
length  succeeded. 

The  reason  why  the  dilatation  of  a  body  is  not  proportional  to  the 
heat,  is,  that  when  this  body  changes  its  state,  its  capacity  for  heat 
also  changes,  so  that  a  greater  or  less  quantity  is  requisite  now  than 
before  the  change,  in  order  to  alter  its  temperature  an  equal  number 
of  degrees.     It  is  true,  we  can  avoid  the  extremes  of  freezing  and 
boiling  at  which  bodies  pass  entirely  from  one  state  to  another  ;  but 
we  cannot  avoid  approaching  these  ;  and  it  is  a  well  known  fact  that 
bodies  are  prepared  for  these  changes  by  imperceptible  degrees,  and 
anticipate,  as  it  were,  the  properties  which  these  changes  develope. 
For  this  reason  doubts  may  be  entertained  whether  the  progressive 
dilatation  of  the  mercury  between  32°  and  212°  is  proportional  to  the 
progressive  increase  of  heat,  although  this  last  point  is  still  very  far 
short  of  that  at  which  mercury  boils.     But  we  may,  without  hesita- 
tion, admit  the  existence  of  this  proportionality  with  respect  to  air 
and  other  aeriform  bodies  which  cannot  be  made  to  change   their 
state  by  any  physical  means.     Accordingly,  by  observing  in  a  great 
number  of  experiments,  the  range  of  the  mercurial  and  air  thermome- 
ters, Gay-Lassac  found  that  it  is  exactly  the  same  in  both  ;  so  that 
between  these  extremes  the  mercurial  thermometer  may  be  consider- 
ed as  indicating  with  great  accuracy  a  corresponding  increase  of  heat. 
Since  that  time  MM.  Petit  and  Dulong  have  ascertained  that  this  pro- 
perty results  from  the  inequalities  of  dilatation  of  the  mercury,  and 
of  the  glass  which  contains  it.     They  found  that  each  of  these  dilata- 
tions, measured  separately,  and  in  an  absolute  manner,  increases  in 
proportion  as  the  temperature  rises,  when  we  compare  it  with  that 
of  dry  air.     But  the  mercurial  thermometer  only  shows  the  differ- 
ence of  these  increments,  which  is  insensible  between  32°  and  212°  ; 
and  hence  it  is  that  these  indications  appear  to  be  perfectly  exact 
within  these  limits.     Nevertheless  they  are  not  so,  and  the  error  in- 
creases with  the  temperature,  but  always  in  consequence  of  the  dif- 
ference of  dilatation  of  the  mercury  and  glass  ;  so  that  when  the 
mercurial  thermometer,  for  example,  indicates  572°,  the  air  thermom- 
eter indicates  only  559°  ,4,  which  shows  an  error  of  only  12,6°. 


Pyrometer.  t  63 

metals.  All  these  instruments  are  yet  very  imperfect.  The  best  is 
the  one  invented  by  Wedgewood.  Its  principle  is  as  follows.  Pure 
clay  and  all  pottery,  the  chief  ingredient  of  which  is  clay,  form  an 
apparent  exception  to  the  law  of  the  dilatation  of  bodies  by  heat. 
Fragments  of  clay  which  have  not  been  baked,  but  only  dried  in 
the  air,  contract  under  the  influence  of  heat,  and  in  proportion  to  its 
intensity ;  and  when  they  are  cooled,  they  do  not  recover  their 
former  dimensions.  The  reason  is,  that  dried  clay  still  contains  a 
certain  quantity  of  water  which  is  gradually  expelled  by  heat. 
Having  observed  this,  Wedgewood  prepared  tubes  of  clay  of  pre- 
cisely determinate  dimensions,  and  then  exposed  them  to  the  action 
of  the  heat  which  he  wished  to  measure.  For  instance,  he  placed 
them  in  a  crucible  with  silver  in  fusion  ;  after  remaining  there  for 
some  time  he  took  them  out,  and  by  means  of  a  very  simple  appa- 
ratus measured  the  diminution  of  their  diameter.  Hence  he  de- 
duced the  degree  of  heat.  To  determine  this  degree,  he  made  use 
of  a  particular  scale,  but  one  which  may  be  easily  compared 
with  that  of  Deluc  or  Fahrenheit.  New.  experiments,  however,  have 
shown  that  the  indications,  given  by  this  instrument,  are  very  uncer- 
tain.* 


*  I  have  myself  suggested  a  method  which  appears  to  be  very  ex- 
act, for  measuring  the  highest  temperatures.  It  is  founded  upon  the 
following  property  which  I  have  demonstrated  from  experiment ;  viz. 
when  a  metallic  bar  exposed  in  a  tranquil  air,  is  plunged  at  one  of 
its  extremities  into  a  source  of  constant  temperature,  the  elevations 
of  temperature  at  each  point  decrease  in  geometrical  progression, 
when  the  distances  from  the  focus  are  in  arithmetical  progression. 
Accordingly,  when  we  have  ascertained  by  experiment  the  propaga- 
tion of  heat  in  a  bar,  it  is  sufficient  to  observe  the  temperature  at 
one  of  its  points,  and  the  distant  e  of  this  point  from  the  constant 
source  of  heat,  in  order  to  determine  the  temperature  of  this  last.  I 
have  employed  this  method  in  determining  the  temperature  of  melt- 
ing tin  and  lead.  The  last  I  found  to  be  375°,,*).  The  diminution 
of  heat  with  the  distance  is  so  rapid,  that  it  would  be  impossible  by 
heating  one  extremity  of  a  bar  of  iron  six  feet  long  and  1  inch 
square,  to  raise  the  temperature  of  the  extremity  one  degree ;  for 
the  heat  necessary  for  this  would  be  so  great  as  to  melt  the  iron. 


64  Heat. 

Some  Remarkable  Points  in  Thermometric  and  Pyrometric  Scales. 


Degrees  of  the  Thermometer. 

Deluc. 

Fahrenheit. 

Wedge- 
wood. 

—      32 

—          40 

A  mixture  of  equal  parts  of  snow  ) 
and  ammonia                             5 

-      14| 

4-       o 

=F           0 
4-        32 

Deep  caverns,  mild  heat  of  spring 
Moderate  heat  of  summer     .     .     . 
Phosphorus  takes  fire      .... 
Temperature  of  human  blood     .     . 
Wax  melts        

4-       10 

+       14 

4-      20 
4-      30 

4-      48 

4-        54 
-j-        « 

+       1» 

4-        99 
4-      HO 

+      63 

4-       174 

4-      90 

4-      234 

4-     164 

-f      400 

4-    190 

4-      460 

4-    209 

-4-       502 

4-    252 

4-       600 

Iron  appears  red  by  day  .  .  .  . 

-j-    464 

4-  2024 

4-     1077 
4-    4587 

0 

27 

4-  2082 

4-    4717 

28 

Gold  melts 

-|-  2315 

4-    5237 

32 

Heat  necessary  to  weld  bars  of  iron 
Extreme  heat  of  a  forge  .  .  .  J 
Cast  iron  melts  

-|-  5953 

4-  7687 
-f-  7976 

4-  13427 
4.  17327 
4-  17977 

95 
125 
130 

Since  the  highest  degrees  of  the  thermometer  can  be  observed 
in  like  manner  by  the  pyrometer,  it  is  evident  that  the  two  scales 
may  be  compared  together,  although  the  thermometer  cannot  be 
used  above  600°.  Many  other  points  are  mentioned  in  Lambert's 
Pyrometry.* 


*  With  respect  to  the  above  table  it  is  proper  to  remark,  that  the 
temperature  of  the  earth  is  not  the  same  in  every  part,  as  the  author 
would  seem  to  intimate.  In  the  sands  of  the  tropics  the  thermometer 
rises  very  high.  At  the  bottom  of  Joseph's  well  in  Egypt,  200  feet 
deep,  it  stands  at  77°  of  Fahrenheit.  In  the  vaults  of  the  Paris  Ob- 
servatory at  54°.  In  Siberia  there  are  places,  it  is  said,  where 
the  earth  never  thaws,  so  that  the  temperature  of  caves  is  never 
above  32°.  Hence  it  appears  that  the  temperature  of  the  exterior 
strata  of  the  earth  goes  on  diminishing  from  the  equator  toward  the 
poles. 


Pyrometer.  65 

163.  Among  the  instruments  made  use  of  to  measure  heat  there 
is  no  one  which  follows  precisely  the  same  gradations  as  heat  itself, 
and  which  may  be  employed  under  all  temperatures.     Still,  in  order 
to  have  a  general  measure,  at  least  ideally,  we  suppose  a  mercurial 
thermometer,  the  variations  of  which  are  exactly  proportional  to 
those  of  heat,  and  which  will  therefore  serve  for  all  temperatures. 
This  ideal  measure  agrees  sufficiently  well  with  the  real  measure  of 
the  mercurial  thermometer  between  the  boiling  and  freezing  points. 
Above  the  boiling  point,  the  real  gradations  are  more  rapid  ;  below 
the  freezing  point  more  slow.     But  we  may  compare  this  ideal 
measure  with  the  indications  of  the  pyrometer  for  very  high  tempe- 
ratures, and  for  degrees  below  the  freezing  point,  perhaps  a  compari- 
son with  the  alcohol  thermometer  would  be  more  suitable.     It  is 
therefore  possible,  in  fact,  to  measure  all  degrees  of  temperature, 
although  the  estimate  of  the  extremes  of  heat  and  cold  must  be  sub- 
ject to  much  uncertainty.    We  shall  close  this  chapter  by  stating  the 
results  of  some  experiments  made  upon  the  dilatation  of  different 
bodies  by  heat. 

164.  Bodies  of  different  kinds,  whether  solid  or  liquid,  are  unequal- 
ly dilated  by  heat.    The  gradations  of  dilatation  are  even  different  in 
the  same  body,  according  to  the  different  degrees  of  heat  to  which 
we  expose  it.     Indeed,  we  find,  with  a  few  exceptions,  that  bodies 
dilate  more  as  they  approach  the  point  at  which  they  are  to  change 
their  state  of  aggregation.     But  hitherto  the  course  of  dilatation 
has  not  been  sufficiently  observed  in  any  body,  with  the  excep- 
tion perhaps  of  mercury,  to  determine  precisely  the  amount  at  each 
temperature.*     The  usual  practice  has  been  to  fix  the  dilatation 
between  the  freezing  and  boiling  points,  and  from  this  to  deduce  it 
proportionally  for  each  degree  of  the  thermometer.     The  following 
table  contains  the  linear  dilatations  of  several  substances  from  0  to 
80°  of  Deluc,  or  from  32°  to  212°  of  Fahrenheit.  By  linear  dilata- 
tion we  understand  that  which  is  measured  in  the  direction  of  one 
and  the  same  dimension  of  the  heated  body,  that,  for  instance,  which 
takes  place  in  the  length  of  a  rule,  which  at  zero  of  Deluc  or  32°  of 
Fahrenheit,  is  supposed  to  be  equal  to  unity.f 


*  Since  this  was  written,  the  Determination  of  which  the  author 
speaks  has  been  completely  effected  by  MM.  Petit  and  Dulong. 

t  The  dilatations  of  glass  and  solid  metals,  stated  in  this  table, 
are  those  ascertained  by  MM.  Lavoisier  and  La  Place. 

Elem.  9 


Heat. 

*-  ;-.::,• 

Fmesrtver        .        ,       • 


0,00!  8782 
'         '  0,0021730 

,  000.2205 

Soft  forced  iron      •-'••    ••  '*' 

0,00  J  2350 

"  • 


English  flint  glass   .         .         .  '•     •         •  0,0008117 

Pure  gold          ......  0,0014661 

Paris  standard  gold         .....  0,0015515 

Platina      .....    '     -         -  °»0008565 

Lead    .         ......  0,0028484 

Glass  of  Saint  Gobin           .      '.      f*'      •  0,0008909 

165.  Two  skilful  observers,  Dalton  at  Manchester,  and  Gay-Lus- 
sac  at  Paris,  have  lately  made  at  the  same  time  very  accurate  ex- 
periments upon  the  dilatation  of  elastic  fluids,  both  vapours  and  per- 
manent gases  ;  and  each  has  found  that  all  elastic  fluids  under  equal 
pressures,  are  equally  dilated  by  heat.  According  to  Gay-Lussac, 
this  dilatation  from  32°  to  212°  is  0,375  of  the  primitive  volume, 
represented  at  32°  by  unity.  According  to  Dalton  it  is  0,398.  The 
first  appears  to  be  the  most  accurate,  because  it  accords  perfectly 
with  very  accurate  experiments  made  upon  atmospheric  air  before 
this  time,  by  the  celebrated  astronomer,  Mayer.  Hence  we  arc 
authorized  to  conclude  that  the  dilatation  of  the  gases  is  a  simple 
effect  produced  by  heat  alone,  but  that  the  dilatation  of  other  bodies 
is  the  compound  result  of  several  forces.  The  dilatation  of  the 
gases  is  exactly  proportional  to  the  intensity  of  the  heat,  which  af- 
fords ground  for  hoping  that  we  may  one  day  be  able  to  measure 
heat  exactly  by  means  of  this  property. 

The  preceding  table  expresses  only  the  dilatation  of  bodies  in  one 
single  dimension  ;  if  we  would  have  the  dilatation  of  the  volume  we 
must  triple  the  numbers  expressed  in  the  table. 

For  example,  the  dilatation  of  mercury  from  32°  to  212°  of  Fah- 
renheit is  expressed  by  0,0061591  ;  taking  the  one  hundred  and  eight- 
ieth part  of  this  we  have  the  linear  dilatation  for  a  degree  of  Fahren- 
heit's scale,  equal  to  0,00003422  of  the  primitive  length  at  32°  ;  trip- 
ling this  number,  we  have  the  cubic  dilatation  equal  to  0,00010266, 
or  a  ten  thousandth  nearly  of  the  primitive  bulk  at  32°,  as  appears 
from  the  experiments  of  MM.  Lavoisier  and  Laplace. 


Inequality  in  the  Dilatation  of  Bodies.  C? 

This  rule  is  founded  upon  a  very  simple  theorem  in  geometry. 
Let  us  suppose  a  homogeneous  volume  V,  which  being  dilated  by 
heat,  becomes  equal  to  W ;  it  will  preserve  a  similar  form  in  these 
two  states.  Now  the  volumes  of  similar  bodies  are  to  each  other  as 
the  cubes  of  their  homologous  sides  ;  for  instance,  as  the  cubes  of 
their  lengths  /,  /',  measured  in  the  same  direction.  We  have,  there- 
fore, tlie  equation 

V       I'3 
^V  ==  I»  ' 

whence 

V  —  V  __  I'3  — I3  _  (l>*  +  II  +  l>)(l>  —  I) 
V  ~  I3  I3 

If  the  linear  dilatation  I'  —  /is  very  small  compared  with  /,  the  dila- 
tation V  —  V  of  the  bulk  will  also  be  very  small  compared  with 
F";  thus  considering  these  dilatations  so  small  that  we  may  without 
material  error  confine  ourselves  to  the  first  power  of  the  fractions 
which  represent  them,  we  shall  see  that  in  the  factor  //2  -+-  / 1'  -f-  /% 
we  may  neglect  them,  and  suppose  Z  =  /' ;  but  then  this  expression 
becomes  3  /2,  and  the  numerator  as  well  as  the  denominator  of  the 
second  member  becomes  divisible  by  /2  ;  performing  this  division, 
we  have 

V  —  V  _  3  (y  —  /) . 


that  is,  by  tripling  the  linear  dilation  — - — ,  which  is  given  by  the  ta- 
ble, we  have  the  dilatation  of  the  bulk  — ^ — ,  as  before  stated. 


Addition. 

Many  phenomena  relative  to  the  dilatation  of  bodies  which  were 
wanting  at  the  time  the  author  composed  this  work,  have  since  been 
determined  with  great  precision  by  MM.  Petit  and  Dulong.  Some 
of  the  most  important  are  the  following. 

(1.)  By  comparing  the  absolute  dilatation  of  dry  air  with  the 
apparent  dilatation  of  mercury  in  glass,  these  philosophers  found 
the  latter  to  increase,  though  by  an  inconsiderable  quantity.  The 
mercurial  thermometer  indicated  300°  (centesimal),  when  the  air 


68  Heat. 

thermometer,  corrected  for  the  dilatation  of  the  glass,  indicated 
292,70.  At  the  boiling  point  of  mercury  the  first  indicated  360°, 
the  second  350°.  The  error,  therefore,  was  only  10°.  The  two 
scales  began  at  the  freezing  point  of  water. 

This  small  difference  was  owing,  as  before  observed,  to  a  compen- 
sation produced  by  the  likewise  increasing  dilatation  of  the  glass  en- 
velope ;  for  the  authors,  having  by  a  particular  method  observed  the 
absolute  dilatation  of  mercury,  compared  with  that  of  air,  found 
that  it  increased  more  rapidly  ;  indeed  from  0  to  100  it  was  T^T7  of 
the  volume  for  a  centesimal  degree  ;  from  0  to  200  3sVi»  ^rom  0 
to  300  1¥Yo-'  The  temperatures  were  estimated  by  the  dilatation 
of  air,  so  that  a  thermometer  founded  upon  the  absolute  dilatation 
of  mercury,  would  have  indicated  at  these  different  temperatures 
0 ;  100°  ;  204°,61  j  314°,15  ;  an'd  thus  for  the  last  there  would  have 
been  an  error  of  14°,15,  which  would  have  been  reduced  to  only 
1 0°,  if  we  had  made  use  of  the  apparent  dilatation  in  the  glass. 

MM.  Petit  and  Dulong  having  also  compared  the  dilatation  of  air 
with  that  of  solids,  found  the  latter  to  be  likewise  increasing,  even  at 
temperatures  very  far  from  that  of  fusion.  Thus  at  the  points  100°, 
200°,  300°,  measured  by  the  absolute  dilatation  of  air,  a  thermome- 
ter, constructed  according  to  the  dilatation  of  a  plate  of  glass,  sup- 
posed to  be  uniform,  will  indicate  100°,  2I3°,2,  352°,9  ;  with  iron 
we  should  have  for  the  two  extremes  1 00°,  372°,6  ;  with  copper 
100°,  328°,8j  with  platina  l(X)o,  31lo,6.  These  results,  which 
are  very  remarkable,  are  sufficient  to  enable  us  to  correct  the  tem- 
peratures which  had  been  calculated  according  to  the  supposition, 
till  then  very  probable,  of  a  dilatation  sensibly  uniform. — See  An- 
nales  de  Chimie,  for  1818. 


CHAPTER  XVIII. 

Changes  produced  by  Heat  in  the  State  of  Aggregation  of  Bodies. 

166.  ONE  very  remarkable  effect  of  caloric  is  the  change  in  the 
state  of  aggregation  which  it  occasions  in  many  bodies.     We  shall 
consider  several  bodies  under  this  point  of  view. 

167.  Water  is  liquid  as  long  as  its  temperature  is  between  32o  and 
212°  of  Fahrenheit.   Being  reduced  to  32°  it  takes  a  solid  state  and 


Effect  of  Heat  upon  the  State  of  Aggregation.  60 

becomes  ice.  During  the  cooling,  its  dilatation  diminishes  till  it  is  at 
about  39°,  when  it  is  at  its  greatest  density.  Below  this  point  it  dilates 
anew,  and  at  32°  it  fills  nearly  the  same  space  it  did  at  46°.  But 
at  the  instant  it  becomes  ice  it  undergoes  a  much  greater  dilatation, 
which  even  acts  with  such  a  force  as  often  to  break  the  most  solid 
vessels.  After  congelation  the  dilatation  still  increases,  until  the  ice 
is  about  ^  rarer  than  water ;  afterwards  it  contracts  with  an  increase 
of  cold  like  all  other  solid  bodies. 

168.  When  water  is  heated  gradually,  its  dilatation  increases  in 
proportion  to  the  intensity  of  the  heat.     When  it  reaches  212°  its 
volume  is  about  ^  greater  than  at  32°,  but  at  212°  bubbles  begin 
to  rise,  and  a  particular  motion  takes  place,  called  ebullition  or  boil- 
ing.    By  making  the  experiment  in  an  apparatus  for  distilling,  we 
find  that  the  bubbles  which  rise  are  not  formed  of  air,  but  of  water 
rendered  elastic,  which  resumes  its  liquid  state  when  cooled.     Its 
bulk  is  so  much  augmented  by  passing  into  the  elastic  state,  that  a 
cubic  inch  of  water  is  thus  made  to  fill  the  space  of  a  cubic  foot ; 
that  is,  it  is  dilated  about  1728  times.     Hence  we  may  easily  con- 
ceive of  the  prodigious  effects  produced  by  steam  in  the  steam  en- 
gine, tzolipyle,  &c. 

169.  The  dilatation  of  mercury  varies  also  by  cooling,  but  in  a 
manner  much  more  uniform  than  that  of  water ;  there  is  no  sensible 
change  before  congelation,  which  takes  place  at  about  —  40°  of 
Fahrenheit ;  but  we  observe  a  very  great  contraction  at  the  instant 
the  mercury  assumes  the  solid  state.     If  it  is  heated  to  600°  it  be- 
gins to  take  the  elastic  form  ;  that  is,  it  enters  into  the  state  of  ebul- 
lition.* 

170.  Very  pure  alcohol  begins  to  boil  at  ,176°.     When  mixed 
with  water  it  supports  a  much  greater  heat  before  changing  its  state. 
Hence  in  Reaumur's  thermometer  the  alcohol  must  be  considerably 
diluted  with  water,  and  yet  the  point  of  ebullition  is  always  too  low 
by  several  degrees.  Before  ebullition  the  alcohol  expands  with  an  in- 
creasing force,  and  its  vapour  possesses  a  high  degree  of  elasticity. f 
The  dilatation  of  alcohol  diminishes  by  cooling,  and  perhaps  for  con- 
siderable degrees  of  cold,  it  has  a  range  more  exactly  conformable  to 

*  From  32°  to  212°  the  absolute  dilatation  of  mercury  is  0,018477. 
t  The  absolute  dilatation  of  the  purest  alcohol  is  |  from  32°  to 

212°. 


70 


Heat. 


that  of  heat  than  mercury.  We  know  no  degree  of  cold  at  which 
it  becomes  solid.*  For  this  reason  the  alcohol  thermometer  is  better 
adapted  to  measuring  extreme  degrees  of  cold  than  the  mercurial 
thermometer. 

171.  Heat  produces  the  same  phenomena  in  many  other  bodies. 
All  fusible  metals  become  liquid  at  a  determinate  degree  of  heat,  and 
elastic  at  one  still  higher.     The  same  is  true  of  all  fusible  bodies. 
But  in  many  bodies  the  passage  from  the  solid  to  the  liquid  state 
does  not  take  place  immediately.     The  gross  oils,  for  example,  can- 
not pass  to  the  state  of  elastic  vapour,  without  some  change  in  their 
chemical  constitution.     There  are  also  solid  bodies  upon  which  the 
highest  degree  of  heat  produces  no  effect,  and  elastic  fluids  of  which 
the  greatest  cold  cannot  change  the  state  of  aggregation.     It  is  on 
this  account  that  we   distinguish   elastic  vapours  from  permanent 
gases.     This  distinction  however,  is  not  very  essential. 

172.  It  remains  for  us  to  speak  of  a  very  remarkable  phenomenon 
produced  by  a  change  in  the  state  of  aggregation.     When  we  mix  a 
pound  of  water  at  167°  with  a  pound  at  32°,  the  result  is  two  pounds 

at  167°  +  S2_  —  99^0.     But  if  we  pour  a  pound  of  water  at  167° 

upon  a  pound  of  ice  at  32°,  we  obtain  two  pounds  of  water  at  the 
temperature  of  32°.  The  whole  heat,  therefore,  of  the  water  used 
is  employed  in  melting  the  ice,  without  raising  its  temperature  at  all. 
We  call  this  heat,  which  thus  eludes  the  senses  and  the  thermome- 
ter, latent  heat  or  combined  caloric,  because  we  consider  the  liquid 
water  as  an  intimate  combination  of  caloric  with  the  matter  of  ice. 

173.  So  far  as  observations  extend  it  appears  that  a  similar  phe- 
nomenon always  takes  place  when  a  body  melts  by  the  simple  effect 
of  heat.     Hence  the  doctrine  that  this  change  always  takes  place  at 
a  determinate  temperature,   which   remains  invariable  during   the 
change,  because  the  heat  which  is  added  is  all  taken  up  in  melting 
the  body. 

174.  When  water  passes  to  the  elastic  state  at  212°,  experience 
proves  that  no  heat  can  increase  its  temperature  ;  and  even  the  va- 
pour which  rises  above  the  water  does  not  indicate,  so  long  as  it  is 
free,  a  temperature  above  that  of  ebullition,  although  this  vapour 
might  be  heated  to  a  much  greater  degree  if  it  were  confined.   Here 

*  [Mr  Button  states  that  he  succeeded  in  solidifying  it  at  a  tempe- 
rature of—  110°.— See  Edinburgh  Encyclopedia,  *rt.  Cold.] 


Effect  of  Heat  upon  the  State  of  Aggregation.  71 

then  it  is  evident  that  there  must  be  combined  caloric,  and  that  all 
the  caloric  which  is  added  is  employed  in  changing  the  water 
into  an  elastic  fluid  ;  consequently,  while  the  change  is  going  on 
there  can  be  no  increase  of  temperature.  The  quantity  of  heat 
which  disappears  or  is  combined  in  this  case,  is  so  great,  that,  accord- 
ing to  the  experiments  of  Watt,  a  temperature  of  975°  would  be  pro- 
duced, if  the  vapour  were  to  return  to  the  state  of  water.  Still  the 
point  of  boiling  water  cannot  be  fixed  at  any  perfectly  constant  tem- 
perature, because  it  varies  with  the  pressure  of  the  air.  The  more 
the  water  is  compressed,  the  more  it  may  be  heated  before  boiling. 
Accordingly,  in  Papin's  Digester,  it  takes  a  heat  much  higher  than 
212°.  On  the  contrary,  under  the  receiver  of  an  air  pump  accu- 
rately exhausted,  water  boils  at  the  temperature  of  about  167°. 
The  boiling  point  of  the  thermometer  should  be  determined  accord- 
ing to  some  fixed  state  of  the  barometer,  as  when  the  mercury  is  at 
30  inches,  for  example. 

175.  According  to  the  best  observations  precisely  similar  phe- 
nomena take  place  in  the  ebullition  of  all  other  fluids.     Hence  the 
following  is  laid  down  as  a  general  law.     Jit  the  instant  of  passing 
either  from  the  solid  to  the  liquid,  or  from  the  liquid  to  the  aeriform 
state,  a  certain  quantity  of  heat  disappears  so  far  as  the  senses  or  the 
thermometer  is  concerned  ;  that  is,  it  becomes  combined. 

176.  In  returning  from  the  aeriform  to  the  liquid,  or  from  the 
liquid  to  the  solid  state,  the  heat  which  had  before  disappeared,  re- 
appears and  becomes  free.     This  is  especially  observable  in  the 
following  phenomenon,  which  sometimes  takes  place  during  the  con- 
gelation of  water.  Fahrenheit  first  observed  that  tranquil  water  may 
be  cooled  considerably  below  the  point  of  freezing  without  ceasing 
to  be  liquid ;  and  more  recent  observations  have  proved  that  it  will 
sometimes  remain  in  this  state  even  at  5°.     But  if  we  disturb  it,  a 
part  immediately  takes  the  form  of  ice,  and  a  thermometer,  plunged 
into  the  fluid,  immediately  mounts  up  to  32°.     This  is  evidently  a 
consequence  of  the  action  of  the  combined  heat,  which  becomes  free 
at  the  moment  the  water  takes  the  solid  state.     Thus  the  most  com- 
mon phenomena  of  congelation  must  exhibit  the  same  effects,  if  we 
observe  them  with  sufficient  attention. 

177.  In  passing  from  the  elastic  to  the  liquid  state,  the  disengage- 
ment of  caloric  heats  the  vessel  much  more  than  would  be  expected 
from  the  quantity  and  temperature  of  the  vapour  precipitated.     It  is 
thus  that  water  is  heated  in  a  condenser,  and  that  a  considerable  quan- 


72  Heat. 

tity  of  cold  water  is  made  to  boil  by  being  exposed  to  the  effect  of  a 
small  quantity  of  elastic  vapour  rising  from  boiling  water. 

178.  As  similar  phenomena  are  observed  to  take  place  in  other 
fluids,  when  passing  from  a  denser  to  a  rarer  state  of  aggregation,  the 
following  may  be  "regarded  as  a  general  law.    In  passing  from  the 
elastic  to  the  liquid  state,  and  from  this  to  the  solid,  a  certain  quan- 
tity of  heat  is  always  disengaged  and  set  free. 

179.  If  such  experiments  do  not  absolutely  prove  the  existence  of 
a  material  caloric,  it  cannot  be  denied  that  they  render  it  very  proba- 
ble.    This  confirms  also  the  opinion  of  those  chemists  who  consider 
the  liquid  and  elastic  fluids  as  chemical  combinations  of  a  solid  sub- 
stance with  certain  quantities  of  caloric. 

180.  We  have  already  remarked,  that  the  state  of  aggregation  of 
a  body  does  not  depend  solely  upon  the  free  heat  which  acts  upon 
it,  but  also  upon  its  chemical  combination  with  other  substances. 
The  phenomena  of  the  absorption  and  disengagement  of  caloric,  ex- 
plained above,  depend  also  upon  the  nature  of  the  combinations  and 
the  state  in  which  they  are  found.     Among  many  experiments  re- 
lating to  this  subject  we  state  the  following. 

When  we  moisten  the  ball  of  a  thermometer  with  ether,  the  liquor 
of  the  thermometer  descends  during  the  evaporation.  When  we 
pour  water  upon  quick  lime  a  part  of  the  water  becomes  solid,  and 
the  mixture  is  heated  considerably.  When  we  dissolve  in  warm 
water  as  much  sulphate  of  soda  as  it  can  dissolve,  and  expose  this 
solution  to  a  very  great  cold,  it  continues  clear  and  fluid  as  long  as 
it  is  at  rest ;  but  if  we  throw  into  this  solution,  already  much  cooled, 
a  crystal  of  the  sulphate  of  soda,  or  if  we  only  agitate  it,  a  certain 
portion  of  the  salt  becomes  instantaneously  crystallized,  and  a  ther- 
mometer, immersed  in  the  fluid,  rises  several  degrees. 

We  shall  see  in  the  following  chapter,  why  the  changes  of  tempe- 
rature, produced  by  chemical  combinations,  are  not  always  governed 
by  the  principles  stated  in  articles  174  and  177. 


Propagation  of  Heat.  73 


CHAPTER  XIX. 

The  Propagation  of  Heat. 

181.  WHEN  bodies  in  which  the  thermometer  indicates  unequal 
degrees  of  heat  are  brought  into  contact,  a  transmission  of  heat  takes 
place  from  die  warmer  to  the  colder,  till  the  thermometer  indicates 
the  same  in  both. 

182.  This  communication  cannot  in  any  way  be  prevented.   Heat 
is,  therefore,  something  which  cannot  be  hindered  from  penetrating 
bodies.     Still  it  propagates  itself  more  readily  and  more  rapidly  in 
some  bodies  than  in  others.     The  best  conductors  of  heat  are  the 
metals  and  water.     The  worst  are  earthy  substances,  ashes,  wood, 
charcoal,  wool,  linen  cloth,  furs,  &c.     The  conducting  power  of  a 
body  may  be  estimated  by  strongly  heating  one  of  its  extremities  and 
observing  the  time  which  it  takes  for  the  heat  to  propagate  itself  to 
the  other. 

183.  In  atmospheric  air  there  are  two  kinds  of  propagation.   The 
first  is  the  one  just  described,  and  considered  in  this  view  air  must 
be  classed  with  the  bad  conductors.     By  the  other  kind  of  propaga- 
tion, heat  passes  off  instantaneously  in  straight  lines  from  the  heated 
body,  and  appears  only  to  traverse  the  air  without  combining  with  it. 
This  is  denominated  radiant  heat.     Scheele  first  observed  it  acci- 
dentally before  the  open  door  of  an  oven.     Many  bodies  reflect  it 
after  the  manner  of  rays  of  light,  particularly  the  metals,  in  so  much 
that  it  may  be  concentrated  at  the  focus  of  a  metallic  mirror.    Other 
bodies  absorb  it  entirely  or  partially.     In  exact  experiments  upon 
heat  this  distinction  in  the  modes  of  propagation  must  be  carefully 
attended  to. 

184.  One  of  the  most  important  discoveries  respecting  heat  is 
due  to  Wilke,  a  Swedish  philosopher,  who  in  the  year  1772  prov- 
ed, that  different  substances  which  indicate  equal  temperatures  by 
the  thermometer,  may  nevertheless  contain  very  unequal  quantities 
of  heat.     The  experiments  which  led  him  to  this  result  were  made 
in  the  following  manner.     If  we  mix  a  pound  of  water  at  32°  with 
a  pound  of  water  at  some  other  temperature,  as  113°,  we  obtain  a 
a  mixture  at  73°,  consequently  at  the  mean  temperature.     But  if 
we  immerse  a  pound  of  metal  at  113°,  in  a  pound  of  water  at  32°, 

Elem.  10 


74 


Heat. 


we  find,  when  the  equilibrium  is  established,  a  much  lower  tempera- 
ture. If,  for  example,  the  body  immersed  be  a  mass  of  iron,  the  wa- 
ter and  iron  will  have  only  a  temperature  of  41°.  Now  if  great  pre- 
caution have  been  taken  to  prevent  the  heat  from  escaping  or  pene- 
trating the  sides  of  the  vessel,  it  is  evident  that  the  water  must  receive 
just  as  much  heat  as  the  iron  loses ;  accordingly  this  quantity  of  heat, 
the  loss  of  which  has  lowered  the  temperature  of  the  iron  72°,  has 
raised  that  of  the  water  only  9°  ;  whence  it  follows  that  8  times  as 
much  heat  is  required  to  change  the  temperature  of  water  one  de- 
gree, as  is  required  to  produce  the  same  effect  in  an  equal  mass  of 
iron. 

185.  The  quantity  of  heat  required  to  change,  one  degree,  the  tem- 
perature of  a  determinate  weight  of  a  body,  is  called  its  specific  heat, 
or  its  capacity  for  caloric.  This  property  may  be  estimated  by  ex- 
periments similar  to  the  above.  If  we  take  for  unity  the  quantity  of 
heat  necessary  to  change  one  degree  the  temperature  of  a  pound  of 
water,  it  is  obvious  that  the  specijic  heat  of  a  body  may  be  represented 
by  a  fraction,  of  which  the  numerator  is  the  number  of  degrees  by 
which  the  temperature  of  water  has  changed,  and  the  denominator  the 
number  of  degrees  by  which  the  temperature  of  the  immersed  body  has 
changed,  the  masses  being  equal.  Thus  in  the  experiment  above 
described,  the  specific  heat  of  the  body  would  be 
W  =  1  =  0,125.* 


*  If  we  call  1  the  specific  heat  of  water,  that  is,  the  quantity  of 
heat  necessary  to  change  one  degree  the  temperature  of  a  pound  of 
water,  a  will  represent  the  heat  necessary  to  change  this  tempera- 
ture a  degrees.  Let  x  be  the  specific  heat  of  the  immersed  body, 
that  is,  the  quantity  of  heat  necessary  to  change  its  temperature  1° ; 
then  6  z  will  be  the  quantity  necessary  to  change  it  b  degrees.  But  in 
our  experiment  the  heat  which  the  water  receives,  is  just  as  great  as 
that  which  the  immersed  body  loses.  We  have  then  a=  bx;  whence 

g 

X  =  6*  If  the  weignt  °f  tne  water  is  not  equal  to  that  which  is 
taken  for  unity,  hut  equal  to  A,  and  that  of  the  body  immersed 
equal  to  B,  we  have  by  similar  reasoning,  x  =  ^ ;  for  A  a  is  the 

quantity  of  heat  acquired  by  the  water,  and  JB  6  z  the  quantity  lost  by 
the  body,  and  these  are  equal. 


Calorimeter.  75 


The  Calorimeter. 

186.  Wilke,  Black,  Crawford,  and  many  other  philosophers,  have 
determined,  in  this  way,  the  specific  heat  of  a  variety  of  bodies. 
There  are,  however,  many  circumstances  in  which  this  method  can- 
not be  employed ;  moreover,  its  results  are  rather  uncertain,  since 
the  conductibility  of  vessels  and  of  the  air,  render  precise  experi- 
ments  almost   impossible.      Lavoisier   and    La  Place,   therefore, 
rendered  a  great  service  to  science  by  inventing  the  calorimeter. 
A  complete  description  of  this  instrument  may  be  found  in  the  Anti- 
phlogistic System  of  Chemistry  by  Lavoisier.   We  shall  only  observe 
in  this  place,  that  the  instrument  is  constructed  upon  the  principle 
that  a  determinate  quantity  of  heat  is  necessary  to  melt  a  determinate 
mass  of  ice.     By  means  of  the  calorimeter  we  are  enabled  to  mea- 
sure the  heat  which  a  body  contains  above  32°,  or  that  which  is 
developed  by  any  chemical  process,  since  we  can  find  with  exact- 
ness how  much  ice  this  heat  is  capable  of  melting.     For  this  pur- 
pose, we  place  the  body  to  be  examined  in  a  space  filled  on  every 
side  with  ice  broken  into  small  pieces,  and  brought  to  the  tempera- 
ture of  32°  by  exposure  for  some  time  in  a  free  air,  supposed  to  be 
above  this  temperature.     We  leave  the  body  thus  enclosed  till  its 
own  temperature  becomes  32° ;  then  we  collect  with  care  all  the 
water  which  has  become  liquid,  and  the  weight  of  this  water  gives 
the  measure  of  the  heat  employed  for  its  liquefaction. 

187.  When  we  wish  to   determine,  with  the   calorimeter,  the 
specific  heat  of  a  body,  it  is  done  by  the  following  very  simple 
process.      We   put  into   the  calorimeter  a   determinate  weight  of 
this  body,  at  a  known  temperature,  1 32°  for  example,  and  suffer 
it  to  melt  as  much  ice  as  it  is  capable  of  melting.    If  we  suppose  this 
to  amount  to  ^  of  a  pound  of  water,  there  has  been  as  much  heat 
employed  as  would  be  necessary  to  raise  T'T  of  a  pound  of  water  from 
32°  to  167°  (172),  or  to  change  1°  the  temperature  of  135  times 
as  much  water ;  that  is,  y/  or  9  pounds.     Now,  as  we  have  repre- 
sented by  1  the  heat  required  to  change  the  temperature  of  a  unit  of 
weight  of  water  one  degree,  the  whole  quantity  of  heat  above  32° 
which  the  body  had  before  the  experiment  must  be  represented  by 
W  or  9.     But  this  heat  had  raised  the  temperature  of  the  body  in 
question  100°  ;  consequently,  the  100th  part  would  be  necessary 
to  change  it  1°  ;  that  is,  its  specific  heat  is  T^  =  T'T.     Finally,  to 


76  HeaL 

comprehend  this  in  a  few  words,  it  is  necessary  to  take  135  times  the 
weight  of  the  melted  ice,  and  divide  the  product  by  the  number  of 
degrees  above  32°  to  which  the  body  was  raised  before  the  experiment, 
in  order  to  ascertain  the  specific  heat  which  belongs  to  it.* 

188.  The  calorimeter  has  over  the  method  of  mixtures  the  es- 
sential advantage  of  giving  the  specific  heat  of  all  solid  and  liquid 
substances  without  exception.     The  inventors  attempted  to  apply 
it  also  to  aeriform  substances ;  but  here  the  results  of  all  methods 
must  be  very  uncertain. 

The  utility  of  the  calorimeter  is  not  confined  to  researches  of  this 
kind.  It  may  be  employed  in  measuring  the  quantity  of  heat  which 
is  disengaged  or  absorbed  in  chemical  combinations.  Our  limits 
however  will  not  allow  us  to  describe  the  method  of  making  these 
experiments. 

189.  We  proceed  to  state  the  specific  heat  of  several  bodies,  as 
determined  by  Lavoisier  and  La  Place  by  means  of  this  instrument. 

1.  Common  water      .       -  ,--.v     >     •  .  -.     .  1, 

2.  Tin       -,. 0,1100 

3.  Crown  glass,  or  glass  without  lead    .         ,  0,1929 

4.  Mercury          .        '..       .  ...      .     .    »         .  0,0290 

5.  Quick-lime  .         ;  /-.••    .  ,     ;  ,•       .     ...   .  0,2169 

6.  Water  and  quick-lime  in  the  ratio  of  9  :  16  0,4391 

7.  Sulphuric  acid  of  the  specific  gravity  of  1,87058  0,3346 

8.  Sulphuric  acid  and  water  in  the  ratio  of  4  :  3  0,6032 

9.  Sulphuric  acid  and  water  in  the  ratio  of  4  :  5  0,6631 

The  following  are  results  given  by  different  observers. 

10-  ^e         V        .        .  "".•    •  .        .         .     0)900 

11.  Mercury       .         ,  .         .         .         .         0)033 

12.  Iron         ...  .         .         .         .     0)125 

13.  Zinc         : •  •.«       .  .         .         .         .         0,067 

14.  Lead       4        .        .  .         .        ...     0,050 

190.  We  shall  now  explain  what  the  above  table  signifies. 

*  The  weight  of  the  body  being  p,  its  temperature  before  the  ex- 
periment <o;  the  weight  of  the  melted  ice  a,  and  the  specific  heat  of 

the  body  x,  we  have  x  =  i^Lf. 
tp 


Calorimeter.  77 

(1.)  If  we  take  for  unity  the  quantity  of  heat  necessary  to  change 
1°  the  temperature  of  a  determinate  weight  of  water,  the  number 
0,125,  for  example,  which  stands  against  iron,  signifies  that  the 
same  weight  of  iron  would  require  only  TVVo>  tnat  ls>  I  °f  this  heat 
to  change  its  temperature  1°. 

(2.)  So  far  as  we  can  suppose  that  the  degrees  of  the  thermome- 
ter increase  and  decrease  in  the  same  proportion  as  the  heat,  we 
may  attribute  another  sense  to  the  numbers  of  the  table.  They 
show  the  ratio  of  the  actual  heat  which  two  bodies  of  equal  weight 
acquire  from  32°  to  the  same  given  temperature.  Thus  the  num- 
bers 0,033  and  0,125,  placed  against  the  numbers  11  and  12,  indi- 
cate that  the  quantities  of  heat  above  32°,  which  mercury  and  iron 
contain  at  the  same  temperature,  are  as  0,033  to  0,125,  or  as  33 
to  125.  If  this  ratio  were  invariable  under  all  temperatures,  these 
numbers  would  express  also  the  ratios  of  the  absolute  quantities  of 
heat.  But  we  cannot  rely  upon  this  result  farther  than  between  the 
temperatures  of  32°  and  212°,  and  we  can  only  apply  it  to  the  quan- 
tity of  heat  over  and  above  that  contained  in  the  body  at  the  tempe- 
rature of  freezing.* 

191.  From  this  it  will  be  easily  seen  in  what  manner  we  calcu- 
late the  quantity  of  heat  which  actually  exists  in  a  given  combination. 
If  we  mix  4  pounds  of  sulphuric  acid  and  5  pounds  of  water  at  the 
same  temperature,  this  mixture  must  contain  besides  its  proper  heat 
at  32°,  4  times  0,3346  +  5  times  1,  that  is  6,3384  of  heat ;  each 
pound,  which  is  the  9th  part  of  the  combination,  will  therefore  contain 
0,7043.  This  number  might  seem  to  represent  the  specific  heat  of 
the  combination  ;  but  according  to  the  estimate  given  in  article  189, 
it  is  only  0,6631.  If  we  examine  other  combinations  in  the  same 
manner,  for  instance,  a  combination  of  water  and  lime,  we  find  a 
similar  deviation. 


*  At  a  temperature  of  t°  above  congelation  every  body  contains 
besides  its  primitive  heat  at  0°,  t  times  as  much  heat  as  is  necessary 
to  change  its  temperature  1°.  Thus,  if  we  multiply  all  the  numbers 
of  the  table  by  t,  we  have  the  excess  of  heat  which  the  body  contains 
at  t°  above  freezing.  But  as  the  ratios  of  numbers  are  not  changed 
when  we  multiply  them  by  the  sa,me  factor,  the  numbers  of  the 
table  may  be  employed  for  this  purpose  at  each  temperature  without 
being  previously  multiplied. 


75 


Heat. 


192.  It  appears,  then,  that  in  such  combinations  of  heterogeneous 
bodies,  internal  changes  of  specific  heat  take  place,  so  as  to  render 
it  impossible  to  determine  a  priori,  by  any  calculation,  the  specific 
heat  of  the  combination.  This  remarkable  phenomenon  is  so  gen- 
eral that  perhaps  no  two  substances  can  be  found  whose  capacity  for 
caloric  does  not  undergo  some  alteration  by  their  chemical  combina- 


tion. 


193.  It  is  by  this  fact  that  we  explain  the  important  phenomenon 
of  the  production  of  heat  or  cold  which  often  appears  in  the  chemical 
combination  of  two  substances.  We  observe  it  in  the  solution  of 
salts.  The  muriate  of  lime,  well  dried,  produces  heat  when  it  is  dis- 
solved in  water ;  crystallized  it  produces  cold.  Calcined  magnesia, 
thrown  into  concentrated  sulphuric  acid  becomes  strongly  heated. 
The  same  is  true  of  lime  or  sulphuric  acid,  when  mixed  with  water. 

Thus,  when  the  heat  which  should  be  contained  in  a  combination, 
according  to  the  estimate  of  article  191,  is  greater  than  the  actual 
specific  heat,  the  inference  is,  that  this  combination  contains  less 
heat  than  its  constituent  parts  would  have  at  the  same  temperature  ; 
that  is,  some  part  of  the  heat  must  have  been  set  free  since  the 
combination.  If,  on  the  contrary,  the  actual  heat  is  greater  than  the 
specific  heat,  deduced  from  calculation,  cold  must  have  been  pro- 
duced by  the  combination. 

This  explains  also  the  reason  why,  in  the  changes  of  the  state  of 
aggregation,  by  chemical  combinations,  the  phenomena  are  not 
always  conformable  to  the  principles  laid  down,  article  175,  &ic.* 

*  By  very  exact  experiments,  made  upon  different  solid  and  liquid 
substances,  MM.  Petit  and  Dulong  have  discovered  that  their  capa- 
cities for  caloric  are  variable  as  well  as  their  dilatations.  The  fol- 
lowing table  contains  their  results. 


Names  of  the  Substances 

Mean  capacities  between 
0°  and  100°  centes. 

Mean  capacities  between 
0°  and  300°. 

Antimony        .     .     . 
Silver     .     .     .    ,.  ... 
Copper  ...  '".     . 

0,0507 
0,0557 
0,0949 
0  1008 

0,0549 
0,0611 
0,1013 
O  1  9*i'> 

Mercury     .     .     . 
Platina  .... 
Glass      
Zinc  

0,0330 
0,0335 
0,1770 

0,0927 

0,0350 
0,0355 
0,0190 
0,1015 

In  these  results  the  capacity  of  water  for  caloric  is  taken  for  unity. 


Production  of  Heat  and  Cold.  79 

CHAPTER  XX. 

Production  of  Heat  and  Cold. 

194.  WE  have  seen  how  heat  and  cold  may  be  produced  by 
means  of  chemical  combinations.     We  remark  also,  that  all  known 
means  of  producing  these  phenomena  are  referred  to  such  combi- 
nations. 

195.  The  best  means  of  obtaining  heat,  as  is  well  known,  is  by 
the  combustion  of  charcoal,  wood,  peat,  &c.     We  should  encroach 
upon  the  province  of  the  chemist  if  we  were  to  attempt  to  ana- 
lyze the  theory  of  combustion.     Still  as  a  knowledge  of  the  best 
means  of  producing  heat  is  important,  we  shall  give  the  following 
short  explanation.     It  has  been  known  for  about  40  years  that  com- 
bustion properly  consists  in  a  chemical  combination  of  bodies  called 
combustible  with  a  certain  constituent  part  of  atmospheric  air,  of 
which  we  have  already  spoken  under  the  name  of  oxygen.     This 
combination  is  accompanied  with  a  disengagement  of  heat  much 
greater  than  takes  place  in  any  other  chemical  combination.     At  the 
moment   the  two  substances  combine,  the  disengagement  of  heat 
is  such  as  to  produce  a  red  colour ;  hence  what  is  denominated 
flame.     The  products  of  this  combination  are  nearly  all  volatile ; 
hence  the  disappearance  of  the  body  burned.     If  the  combustible 
body  is  composed  wholly  or  partly  of  substances  which  become  aeri- 
form at  an  elevated  temperature,  these  parts  rise  under  the  form  of 
vapour  when  the  temperature  has  attained  the  degree  necessary  for 
inflammation;  and  at  the  moment  they  become  red  by  combining 
with  oxygen  they  form  flame. 

196.  Combustion  is  favoured  and  the  heat  increased  by  the  ac- 
tion of  a  strong  current  of  atmospheric  air ;  hence  the  use  of  bel- 
lows.    When  we  employ  a  current  of  oxygen  gas  instead  of  atmo- 
spheric air,  we  have  the  greatest  heat  which  is  known.        v ! 

It  might  perhaps  be  difficult  to  account  for  the  heat  produced  by 
combustion,  if  we  consider  it  merely  as  a  change  of  specific  heat ; 
but  the  uncertainty  which  prevails  respecting  the  specific  heat  of 
gases,  makes  it  impossible  to  form  any  definite  opinion  on  this  subject. 

197.  A  second  very  powerful  means  of  producing  heat  is  by  the 
solar  rays.     Their  action  is  more  intense  ;    1.  According  as  their 


80  Heat. 

direction  is  more  vertical ;  hence  the  difference  in  their  effect  in 
different  parts  of  the  earth  and  at  different  seasons.  2.  According 
as  they  are  more  concentrated  ;  hence  the  excessive  heat  produced 
by  concave  mirrors  and  burning  glasses.  When  these  instruments 
are  of  sufficient  magnitude  their  effects  are  in  no  respect  inferior  to 
the  heat  produced  by  oxygen  gas,  and  they  may  even  surpass  it. 
3.  The  effect  of  the  solar  rays  depends  upon  certain  material  pro- 
perties peculiar  to  each  body.  Thus  it  appears  that  transparent 
bodies  are  not  immediately  heated  by  them ;  and  among  opake 
bodies  those  of  a  light  colour  are  heated  much  less  than  those  of 
a  deep  colour,  especially  black.* 

198.  There  are  also  many  other  circumstances,  not  well  explained, 
attending  this  mode  of  procuring  heat.     Formerly  the  sun  was  re- 
garded as  a  real  mass  of  fire,  and  consequently  its  heat  was  not  sup- 
posed to  differ  essentially  from  the  radiant  heat  of  our  terrestial  fires. 
More  recently  this  hypothesis  has  been  called  in  question,  and  phi- 
losophers have  regarded  the  sun  as  a  body  in  itself  dark,  and  only 
surrounded  by  a  luminous  atmosphere.     They  have  not  allowed  to 
the  solar  rays  any  proper  heat,  but  only  a  force  of  impulsion  capable 
of  exciting  the  caloric  contained  in  bodies.     But  what  we  have  said 
respecting  caloric  seems  to  show  that  an  elevation  of  temperature 
cannot  be  produced  but  by  an  augmentation  of  the  caloric  itself,  or 
a  diminution  of  the  capacity  of  bodies  for  caloric  ;  and  neither  oi 
these  causes  is  consistent  with  the  above  hypothesis.  The  celebrated 
Herschel  very  recently  made  experiments  which  tend  to  prove  thai 
the  sun  not  only  sends  rays  of  light,  but  also  rays  of  a  peculiar  heat, 
the  laws  of  which  do  not  agree  with  those  of  our  radiant  heat.    The 
experiments  of  Herschel  are  very  remarkable  and  deserve  greal 
attention. 

199.  There  is  also  a  third  means  of  obtaining  heat  and  cold, 
founded  upon  the  principle,  that  the  compression  of  bodies  is  attended 
with  a  developement  of  heat,  and  their  dilatation  urith  the  opposite 
effect.     Many  phenomena,  a  long  time  insulated,  are  referred  tc 
this  law.     The  heat  produced  by  friction  is  without  doubt  a  conse- 


*  I  have  seen  in  the  valley  of  Chamouni  among  the  Alps,  a  specie: 
of  schistus  or  blackish  earth,  which  the  inhabitants  of  the  country 
spread  over  the  ground  in  the  spring,  in  case  the  snow  lies  late  01 
comes  unexpectedly.  If  there  is  only  one  or  two  feet  of  snow,  t 
day  is  sufficient  to  melt  it  when  covered  with  this  black  earth. 


Artificial  Cold.  81 

quence  of  compression  thus  occasioned.  .The  intensity  of  the  heat 
is  in  proportion  to  the  degree  of  compression.  It  has  also  for  a 
long  time  been  observed  that  the  condensation  of  air  produces  heat, 
and  its  rarefaction  cold.  An  experiment  recently  made  at  Lyons 
has  shown  that  this  effect  is  much  greater  than  was  formerly  believed, 
since  a  small  mass  of  air  condensed,  by  a  violent  blow,  to  about  j\ 
of  the  space  naturally  occupied  by  it,  developes  so  much  heat,  that 
we  can  not  only  inflame  phosphorus,  but  also  tinder  and  other  com- 
bustible substances. 

200.  Berthollet  has  established  a  principle  by  which  many  phe- 
nomena are  easily  explained.  Heat  produces  an  elevation  of  tempe- 
rature, so  long  as  there  are  any  obstacles  to  oppose  the  dilatation  of 
bodies.  The  phenomena  which  take  place  in  the  changes  of  the 
state  of  aggregation,  the  uniform  dilatation  of  all  the  elastic  fluids,  as 
well  as  the  phenomenon  mentioned  above,  appear  to  be  necessary 
consequences  of  this  principle. 


Artificial  Cold. 

201.  There  are  methods  of  producing  a  greater  or  less  degree  of 
cold ;  but  they  may  all  be  referred  to  the  law  relating  to  the  passage 
of  bodies  from  a  denser  to  a  rarer  state  of  aggregation. 

Among  these  means  we  first  distinguish  evaporation,  which  is 
always  attended  with  a  greater  or  less  depression  of  temperature  ac- 
cording to  its  rapidity.  Ether  and  alcohol,  for  this  reason,  produce 
the  most  powerful  effects.  The  evaporation  of  water  also,  in  phe- 
nomena of  daily  occurrence,  produces  a  considerable  degree  of  cold. 
And  in  the  East  Indies  this  method  is  even  employed  to  procure 
quantities  of  ice. 

The  second  method  is  by  a  dissolution  of  most  of  the  salts  in  wa-. 
ter,  and  probably  of  all,  provided  they  contain  all  their  water  of  crys- 
tallization. This  dissolution  changes  the  salt  from  a  solid  into  a 
fluid.  Most  of  the  salts,  however,  produce  only  a  feeble  degree  of 
cold.  On  the  contrary,  a  salt  which  is  deprived  of  its  water  of  crys- 
tallization gives  heat  instead  of  cold,  because  the  salt  first  attracts 
the  water  and  puts  it  into  a  solid  state  before  being  itself  dissolved. 

The  solution  of  crystallized  salts,  when  it  takes  place  in  sulphuric 
and  nitri*  acid,  produces  a  still  greater  effect,  particularly  that  of  the 
crystals  of  sulphate  of  soda,  which  contain  much  water  in  a  solid 

Elem.  11 


ft  Heat. 

state.*  But  the  most  energetic  effects  are  produced  by  mixtures  of 
crystallized  salts  with  snow  or  pounded  ice,  in  which  the  two 
constituent  parts  pass  at  once  to  the  liquid  state.  No  salt  has  yet 
been  discovered  whose  effects  are  more  powerful  in  this  respect  than 
the  muriate  of  lime,  when  it  contains  all  its  water  of  crystallization. 
As  a  great  quantity  of  this  crystal  may  be  obtained,  it  is  not  now  diffi- 
cult to  freeze  considerable  portions  of  mercury. 


*  Sulphate  of  soda  mixed  with  a  little  nitric  or  sulphuric  acid  con- 
centrated, produces  cold ;  but  when  mixed  with  a  great  quantity  of 
these  acids,  it  produces  heat. 


SECTION  IV. 

LIQUID   BODIES. 


CHAPTER  XXI. 

Liquids  in  General. 

202.  THERE  are  few  substances  which  are  liquid  naturally  and  in 
their  simple  state  ;  nevertheless,  besides  water  and  mercury,  we  may 
always  consider  alcohol,  ether,  and  the  fluid  oils,  as  possessing  this 
property.     But  these  liquids,  and  particularly  water,  have  the  power 
of  dissolving  so  great  a  number  of  solid,  liquid,  and  even  aeriform 
substances,  that  we  find  an  infinite  variety  of  liquids,  if  we  consider 
all  solutions  as  belonging  to  this  class. 

Water. 

203.  The  influence  which  water  exerts,  as  well  upon  inorganic 
as  organic  bodies,  and  the  multiplied  uses  which  are  made  of  it,  can- 
not escape  the  most  inattentive  observer. 

204.  In  its  pure  state  water  is  perfectly  transparent,  without  colour, 
odour,  or  sensible  taste.     But  it  can  only  be  obtained  in  this  state 
by  repeated  distillations.     Nature,  however,  furnishes  it  almost  pure 
in  rain  and  snow.     The  water  of  seas,  rivers,  fountains,  and  wells, 
always  contains  foreign  substances  in  solution,  particularly  saline  sub- 
stances, and  even  organic  substances,  of  which  hitherto  little  no- 
tice has  been  taken.  These  different  modifications  render  the  waters 
in  question  more  or  less  appropriate  to  our  use. 

205.  The  force  of  affinity  which  water  possesses  gives  it  the 
power,  not  only  of  dissolving  many  bodies,  but  also  of  combining 
with  many  solid  and  fluid  substances,  and  with  all  aeriform  suh- 


g4  Liquid  Bodies. 

stances.  In  combinations  of  this  kind  it  often  becomes  solid  or  elas- 
tic, and  then  it  entirely  eludes  our  senses. 

206.  The  action  of  water  upon  a  pure  salt  separated  from  every 
foreign  admixture  is  particularly  worthy  of  notice.     According  as 
the  quantity  of  water  or  salt  preponderates,  the  salt  is  rendered  fluid 
by  the  water,  or  the  water  is  rendered  solid  by  the  salt ;  this  is  the 
reason  why  every  salt  contains  a  certain  quantity  of  solid  water,  call- 
ed water  of  crystallization,  because  it  was  supposed  that  the  salts 
were  incapable  of  crystallization  without  this  water.     Some  salts 
attract  water  so  powerfully  that  they  dissolve  in  the  air  ;  others,  on 
the  contrary,  are  so  easily  deprived  of  their  water  by  the  air  that 
they  become  decomposed  and  crumble   into  powder.     The   first 
species  comprehends  the  dried  salts  which  are  artificially  deprived 
of  their  water  of  crystallization.     The  most  remarkable  of  this  kind 
are  potash  and  muriate  of  lime. 

207.  We  have  shown  in  the  preceding  section  how  water  is  modi- 
fied by  heat.     We  have  seen  also,  that  water  is  not  a  simple  sub- 
stance, as  was  once  supposed,  but  a  compound, of  oxygen  and  hy- 
drogen.    Some,  however,  have  lately  undertaken  to  deny  the  de- 
composition of  water.    It  has  been  pretended  that  water  changes  to  a 
permanent  gas  by  a  mere  combination  with  caloric  or  with  some 
other  imponderable  substance ;  and  that  from  this  state  it  may  be 
reduced  to  water  by  a  contrary  process.     But  this  opinion  rests 
upon  a  very  uncertain  foundation. 

208.  Among  the  mechanical  properties  of  water  its  weight  is  par- 
ticularly important.     [A  cubic  foot  of  water  at  the  temperature  of 
50°,  weighs  1000  avoirdupois  ouncos.     It  is  usual  in  experiments 
upon  specific  gravities  to  refer  to  the  temperature  of  60°  at  which  a 

lb. 

cubic  foot  of  water  weighs  62,353,  or  a  little  less  than  1000  ounces. 
In  the  more  accurate  experiments,  however,  upon  this  subject  the 
absolute  weight  is  first  ascertained,  and  the  cubic  inch  is  taken  as 
the  measure,  this  bulk  of  distilled  water  at  the  temperature  of  60° 
being  estimated  at  252,525  grains.] 

209.  It  has  been  proved  by  experiment  that  compressibility*  dila- 
tability,  and  elasticity  belong  to  water ;  but  it  is  only  when  a  very 
considerable  force  is  applied  that  any  such  effect  is  produced. 

[*  The  phenomena  of  the  transmission  of  sound  through  water  and 
other  liquids  had  long  indicated  that  they  were  capable  of  being 
compressed.  Canton,  an  English  philosopher  clearly  detected  this 


Mercury.  85 

210.  It  is  not  proved  in  a  decisive  manner,  either  by  experiment 
or  theory,  that  water  in  porous.  It  appears  to  our  senses  like  a  per- 
fectly continuous  mass ;  and  experiment  shows  that  it  is  perfectly 
impenetrable  to  the  most  subtle  gases,  provided  no  chemical  affinity 
exerts  its  influence  in  effecting  a  combination.* 


Mercury. 

211.  Since  extensive  use  is  made  of  mercury  in  physical  experi- 
ments some  account  ought  to  be  given  of  this  substance.  It  is  a  true 
and  perfect  metal,  approaching  even  in  its  chemical  properties  to  the 
precious  metals.  In  its  solid  state,  it  has  all  the  appearance  of  a 
metal.  The  purest  is  that  which  is  obtained  from  red  lead.  Its 
specific  gravity  is  13,586,  water  being  unity.  With  respect  to  its 
chemical  properties,  it  is  to  be  observed  that  it  easily  dissolves  all 
metals  except  iron.  Its  combination  with  other  metals  is  called  an 


property  by  observing  the  volume  occupied  respectively  by  oil,  wa- 
ter, and  mercury,  first  placed  in  a  vacuum,  and  afterwards  exposed 
to  the  pressure  of  the  atmosphere  ;  but  the  results  which  he  obtain*- 
ed,  though  exact  in  themselves,  were,  however,  liable  to  be  affected 
by  the  accidental  variations  of  form  and  temperature  to  which  the 
apparatus  was  subject.  M.  Oersted  completely  removed  these  diffi- 
culties by  plunging  the  liquid  to  be  compressed,  together  with  the 
vessel  containing  it,  into  another  liquid  to  which  the  pressure  was 
applied,  and  through  which  it  was  made  to  pass  to  the  interior  liquid 
without  changing  the  form  of  the  vessel,  since  it  acted  equally  within 
and  without.  M.  Oersted  found,  likewise,  that  a  pressure  equal  to 
the  weight  of  the  atmosphere  produces  in  pure  water  a  diminution 
of  volume  equal  to  0,000045  of  its  original  volume.  The  experi- 
ments of  Canton  gave  0,000044.  M.  Oersted  found,  by  varying  the 
pressure  from  $  of  the  weight  of  the  atmosphere  to  6  atmospheres,  a 
change  ef  volume  sensibly  proportional  to  the  pressure.  Later  ex- 
periments, made  by  Mr  Perkins,  seera  to  show  that  this  proportion- 
ality continues  when  the  pressure  amounts  to  2000  atmospheres. 
Before  the  water,  however,  is  entirely  freed  from  air,  the  diminution 
of  volume,  produced  by  the  pressure,  is  at  first  somewhat  greater 
than  the  above  ratio  would  indicate.] 

*  I  believe  it  would  be  impossible  to  find  any  circumstances  or 
disposition  of  apparatus,  in  which  this  difficulty  would  not  exist. 


36  Liquid  Bodies. 

amalgam.  In  making  experiments  it  is  necessary,  on  account  of  this 
property,  to  avoid  bringing  it  in  contact  with  any  metal,  except  iron. 
It 'may  be  kept  in  vessels  of  glass,  earth,  or  wood. 


Alcohol. 

212.  All  soft  juicy  vegetables  are  susceptible  of  vinous  fermenta- 
tion.    While  this  process  is  going  on,  a  part  of  the  saccharine  sub- 
stance which  they  contain,  changes  into  an  inflammable  liquid  called 
alcohol.     By  distilling  this  alcohol  with  a  gentle  heat  it  may  be  sep- 
arated from  the  other  constituent  principles  or  vegetable  juices,  by 
reason  of  its  great  volatility.     It  is  impossible,  however,  to  prevent 
a  considerable  quantity  of  water  from  mixing  with  it.     By  a  new 
distillation  it  becomes  more  purified,  and   is  then  called  aqua  vitte. 
The  distillation  being  repeated  several  times,  it  becomes  more  and 
more  separated  from  water,  and  takes  the  name  of  rectified  alcohol, 
It  cannot,  however,  be  so  far  purified  by  simple  distillations,  as  not 
to  contain  about  {  of  its  weight  of  water.     If  we  wish  to  rectify  it 
farther  it  must  be  distilled  over  a  salt  deprived  of  its  water  of  crys- 
tallization and  perfectly  dry.     It  is  believed  that  this  operation  dis- 
engages it  entirely  from  all  the  water  it  contained,  and  it  is  then 
called  pure  alcohol. 

213.  Pure  alcohol  is  therefore  a  liquid  which  is  derived  immedi- 
ately from  organic  nature.     It  is  perfectly  transparent  and  without 
colour,  of  a  burning  taste  and  agreeable  odour.     It  burns  with  a 
bluish  flame  and  produces  no  smoke,  no  soot,  nor  any  kind  of  resi- 
duum.   Its  specific  gravity,  when  pure,  is  0,792,  water  being  1.     It 
mixes  with  water  in  all  proportions,  and  appears  to  produce  an  inter- 
nal heat.    The  combination  of  the  two  substances  occupies  less 
space  than  the  two  filled  separately.     In  the  third  section,  we  have 
already  considered  it  in  its  relation  to  heat.      It  dissolves  many 
bodies,  particularly  resins,  by  combining  with  which  it  forms  what  is 
called  varnish.  It  dissolves  many  salts  and  has  no  effect  upon  others. 
In  its  pure  state,  alcohol  is  exempt  from  all  fermentation  and  cor- 
ruption, and  preserves  bodies  which  are  immersed  in  it. 


General  Remarks  upon  Liquids,  87 


Ether. 

214.  Ether  is  properly  an  alcohol  in  which  the  proportions  of  the 
constituent  principles  are  changed.  Ordinarily  they  are  prepared  for 
this  new  combination  by  distilling  a  mixture  of  alcohol  and  sulphuric 
acid  with  a  gentle  heat.  The  product  is  then  called  sulphuric  ether. 
It  is  without  colour,  of  a  very  penetrating  and  agreeable  odour,  and 
a  burning  taste ;  it  is  extremely  volatile,  and  becomes  inflamed 
not  only  by  contact  with  a  burning  body  like  alcohol,  but  even  by 
the  near  approach  of  one.  It  burns  in  the  same  manner  as  alcohol, 
but  deposites  a  small  residuum.  It  dissolves  many  bodies,  particu- 
larly the  resins  ;  and  for  that  which  is  called  gum  elastic,  or  India 
rubber,  it  is  the  only  known  solvent  which  does  not  destroy  its  elas- 
ticity. It  is  the  lightest  of  all  known  liquids.  The  specific  gravity  of 
ether,  carefully  prepared,  is  only  0,745  ;  and  according  to  Lowitz,  it 
is  capable  of  being  purified  still  more.  If  we  mix  it  with  nearly  an 
equal  quantity  of  water,  the  two  substances  do  not  form  a  homoge- 
neous combination  ;  but  when  the  mixture  becomes  tranquil,  we  dis- 
tinguish two  liquids,  of  which  the  upper  is  composed  mostly  of  ether 
with  a  very  little  water,  and  the  lower  mostly  of  water  with  a  very- 
little  ether.  It  does  not,  therefore,  combine  with  water  in  all  pro- 
portions. 


General  Remarks  upon  Liquids. 

215.  It  is  proved  by  experiment  that  all  liquids,  with  the  excep- 
tion perhaps  of  the  grosser  oils,  take  the  elastic  state  when  exposed 
in  small  quantities  in  a  vacuum.  From  this  phenomenon  we  neces- 
sarily infer,  that  for  all  liquids  in  which  it  occurs,  the  liquid  state  is 
not  so  much  the  consequence  of  an  internal  attraction,  as  of  an  exter- 
nal pressure,  arising  in  part  from  the  gravity  of  the  liquid  itself,  and  in 
part  from  the  external  air ;  a  hypothesis  which  accords  perfectly 
with  that  of  article  13.  For,  since  the  liquid  state  supposes  an  ab- 
solute equilibrium  between  the  attractive  force  of  the  matter  and  the 
force  of  the  caloric  which  tends  to  dilate  it,  it  is  reasonable  to  sup- 
pose that  throughout  the  natural  world,  where  we  do  not  find  any 
exact  equilibrium,  there  does  not  perhaps  exist  a  body  which  could 


gg  Liquid  Bodies. 

preserve  its  liquid  state  by  means  of  its  internal  forces  alone.  The 
perpetual  instability  of  the  action  of  heat  would  aLo  be  an  obstacle 
to  this  equilibrium.  But,  on  the  contrary,  there  may  be  many  bodies 
in  which  one  or  the  other  only  of  these  forces  preponderates.  Thick 
and  viscous  substances  appear  to  owe  their  state  to  a  feeble  prepon- 
derance of  the  force  of  cohesion.  In  other  substances  the  slight  ex- 
cess of  the  expansive  force  is  kept  in  equilibrium  by  external  pres- 
sure. This  was  essentially  the  opinion  of  Lavoisier  ;  only  he  seems 
to  have  paid  too  little  attention  to  the  proper  gravity  of  the  fluid  mat- 
ter.* 


CHAPTER  XXH. 

The  Specific  Gravity  of  Solids  and  Liquids. 

216.  ALTHOUGH  the  ordinary  methods  of  estimating  specific  grav- 
ity belong  to  hydrostatics,  there  is  one  which  can  be  here  completely 
explained.     It  is  the  method  of  Klaprotb,  and  it  is  the  more  deserv- 
ing of  being  made  known,  inasmuch  as  it  has  never  to  my  know- 
ledge been  described,  though  in  most  cases  it  ought  to  have  the 
preference  over  other  methods,  on  account  of  its  simplicity,  conven- 
ience, and  exactness.     The  whole  apparatus  required  consists  of  a 
correct  balance,  and  one  or  more  glass  vessels  shutting  air  tight. 

217.  To  find  the  specific  gravity  of  a  liquid.     First  weigh  an 
empty  phial ;  that  is,  balance  it  by  weights.     Then  weigh  it  filled 
with  distilled  water,  taking  care  to  stop  it  accurately.     Next  fill  it 
with  the  liquid  to  be  examined,  and  divide  this  weight  by  that  of 


*  All  liquids  vaporize  in-  a  vacuum,  even  the  gross  oils  ;  and  the 
elastic  force  which  they  exhibit  is  greater,  according  as  their  ebulli- 
tion takes  place  at  a  lower  temperature.  Thus  ether,  which  boils  at 
100°,  has  at  the  temperature  of  66°,  a  considerable  elasticity,  which 
becomes  manifest  in  a  vacuum;  water,  which  boils  at  212°  would 
have  at  66°  an  elasticity  much  less ;  and  the  gross  oils  which  boil  only 
at  high  temperatures  would  have  one  still  less  ;  for  at  the  boiling 
temperature  they  only  have  a  force  sufficient  to  counterbalance  the 
weight  of  the  atmosphere  ;  and  it  is  not  strange,  therefore,  that  they 
have  a  very  feeble  one  at  low  temperatures.  We  shall  resume  this 
subject  when  we  come  to  treat  of  evaporation. 


Specific  Gravity  of  Solids  and  Liquids.  89 

the  first ;  the  quotient  will  be  the  specific  gravity  sought.  If,  for  ex- 
ample, the  phial  contain  864  grains  of  distilled  water,  and  only  673 
grains  of  ether,  the  specific  gravity  of  the  evher  is  f|f  —  0,779. 

The  accuracy  of  this  process  is  evident  from  what  has  been  said 
respecting  specific  gravities  (53.) 

218.  ^Tofind  the  specific  gravity  of  solid  bodies  which  do  not  dis- 
solve in  water.  To  estimate  the  specific  gravity  of  a  solid  body,  it 
is  only  necessary  that  the  body  should  be  introduced  into  the  vessel ; 
but  it  is  not  necessary  that  it  be  a  single  fragment ;  it  may  even  con- 
sist of  fine  dust.  We  may,  however,  for  more  bulky  substances  pro- 
cure vessels  with  an  opening  sufficiently  large  to  receive  them.  The 
most  simple  manner  of  performing  the  experiment  is  the  following ; 

First  weigh  the  vessel  exactly  filled  with  distilled  water.  Then 
place  the  body  to  be  examined  in  the  same  scale  with  the  vessel ; 
and  put  in  the  other  scale  a  weight  sufficient  to  balance  it.  We  thus 
ascertain  the  weight  of  the  body.  Then  take  away  the  body  and 
the  vessel,  and  introduce  the  body  into  the  vessel  filled  with  water. 
The  vessel  is  then  to  be  closed,  and  great  care  must  be  taken  that 
no  bubble  of  air  be  left  in  the  interior.  Having  wiped  the  exterior 
dry,  it  is  to  be  placed  in  the  scale  where  it  was  before.  This  will 
then  be  lighter  than  the  other,  and  as  much  weight  must  be  added  to 
it  as  is  necessary  to  establish  the  equilibrium.  This  weight  which  is 
added,  indicates  how  much  water  the  body  displaced  from  the  vessel. 
Divide  the  weight  of  the  body  by  the  weight  of  the  water  displaced, 
and  the  quotient  will  be  the  specific  gravity  sought.  If,  for  example, 
the  body  weighs  523  grains,  and  the  water  displaced  84  grains,  the 
specific  gravity  of  the  body  will  be  =  yT3  —  6,226.* 

*  In  order  that  a  balance  may  be  rigorously  exact,  its  two  arms 
must  be  perfectly  equal,  and  the  same  weight  placed  in  either  scale 
must  equally  produce  an  equilibrium,  the  body  weighed  being  the 
same.  But  it  is  almost  impossible  to  attain  to  such  perfection  ; 
and  if  we  would  take  (he  trouble  to  subject  those  balances  which  pass 
for  the  most  exact  to  an  examination  of  this  kind,  we  should  scarcely 
find  one  that  would  bear  the  test.  But  we  can  dispense  with  this 
degree  of  perfection  ;  and,  provided  the  balance  is  sensible,  deter- 
mine the  weight  of  a  body  as  well  as  if  the  two  arms  were  exactly 
equal.  We  have  only  to  weigh  in  the  same  scale  both  the  body  and 
the  weight  with  which  we  compare  it.  In  the  first  place,  we  put  in 
one  scale  the  body  to  be  weighed,  -and  counterbalance  it  with  some 
heavy  substance  ;  when  the  equilibrium  is  established,  we  take  the 

Elfm.  ix> 


go  Liquid  Bodies. 

219.  There  are  bodies  which  imbibe  water,  without  being  dissolv- 
ed or  decomposed.     With  respect  to  these,  the  determination  of  the 
specific  gravity  seems  to  present  an  ambiguity.     In  case  of  free- 
stone, for  example,  we  may  wish  to  ascertain  its  specific  gravity  ab- 
stracted from  the  interstices  which  it  contains  ;  that  is,  we  may  seek 
what  would  be  the  specific  gravity  of  the  same  weight  and  volume  of 
this  stone,  provided  it  had  no  sensible  interstices.     Or  we  may  seek 
the  specific  gravity  of  the  proper  mass  of  the  body.     In  these  two 
cases  we  proceed  as  follows.     We  first  weigh  the  body  in  the  air. 
Suppose  it  weighs  1000  grains.     We  then  immerse  it  in  water  until 
it  has  imbibed  all  that  it  will,  and  ascertain  how  much  its  weight  is 
augmented.     Suppose  this  augmentation  to  be  50  grains.     We  then 
introduce  this  body  into  the  vessel,  and  see  how  much  water  it  dis- 
places ;  suppose  this  to  be  240  grains.    Now  if  we  wish  to  determine 
the  specific  gravity  in  the  first  sense,  we  divide  1000  by  240,  and  find 
it  to  be  4, 167.  If  we  would  determine  it  in  the  second  sense,  we  must 
consider  that  the  proper  mass  of  the  body  has  not  displaced  240 
grains  of  water,  but  240  —  50  =  190  grains.     Its  specific  gravity 
is  therefore  YW  =  5>263- 

When  bodies  become  decomposed  in  water,  as  is  the  case  with  most 
argillaceous  substances,  this  double  signification  still  holds.  Only  in 
the  first  sense,  when  we  consider  the  body  as  a  continuous  mass,  the 
specific  gravity  must  be  uncertain  in  itself,  since  a  body  of  this  kind 
may  have  a  very  different  density  on  account  of  the  solution  of  the 
continuity  of  its  parts.  In  this  case  we  can  only  form  an  approximate 
estimate.  We  obtain  it  by  the  same  process  as  in  article  218,  ob- 
serving only  to  close  the  vessel  before  the  body  is  decomposed.  If, 
on  the  contrary,  we  wish  to  know  the  specific  gravity  in  the  second 
sense,  we  have  but  to  crumble  the  body  as  fine  as  possible,  and 
then  to  proceed  as  in  article  218. 

220.  To  find  the  specific  gravity  of  bodies  which  dissolve  in  water. 
When  we  wish  to  know  the  specific  gravity  of  a  salt,  or  any  other 
body  which  dissolves  iri*water,  we  make  choice  of  another  liquid,  as 
alcohol  or  oil,  in  which  it  does  not  dissolve.     We  first  determine,  as 
in  article  217,  the  specific  gravity  of  this  liquid  compared  with  water. 

body  away  and  substitute  weights  till  the  equilibrium  is  restored.  It 
is  evident  that  these  weights  represent  exactly  the  weight  of  the  body 
whose  place  they  take,  independently  of  any  inequalities  in  the  bal- 
<""•-  This  method  of  double  weighing  was  suggested  by  Borda. 


Specific  Gravity  of  Solids  and  Liquids.  91 

Suppose  it  be  0,866.     We  then  determine  the  specific  gravity  of  the 
salt  with  reference  to  this  liquid,  as  in  article  218.     Suppose  it  be 
3,278.     Lastly,  we  multiply  these  two  numbers  together,  and  their 
product  2,829718,  expresses  the  specific  gravity  of  the  body.* 
221.  We  shall  here  present  a  short  list  of  specific  gravities. 

Platina         ....  20,722  Klaproth. 

Gold        *.-,"•     .         .         .  19,258  Brisson. 

Mercury       ....  13,586  Fischer. 

Lead       .     ^         .    -     .    <-•:  .  11,352  

Silver  ....  10,784  Klaproth. 

Bismuth 9,070  Brisson. 

Copper         .         .     -'  „   •'    .   4  8,876  

Brass       .    '-•'.       '.         .'        .  8,395  

Iron     .       ;*>*  - •'••••;•        .       •• .  7,800  Bergman. 

Steel  .  7,767  Musschenbroek. 


*  The  specific  gravity  of  water  being  1,  let  that  of  the  liquid  be  a, 
and  that  of  the  body  6.  Let  the  vessel  contain  p  of  the  water  and  q 
of  the  other  liquid.  Let  the  weight  of  the  salt  be  r,  and  that  of  the 
liquid  displaced  by  it  s.  We  shall  have 

1  :a::p:q, 
a :  b   : :r  : s; 

multiplying  the  two  proportions,  term  by  term,  and  omitting  the  com- 
mon factor,  we  obtain 

1  :  6  :  :  p  r  :  q  s. 

Whence  6  =  —  z=  -  X  ->  which  is  the  rule  given  above. 
pr      p       r 

In  all  the  preceding  operations  it  is  necessary,  if  we  would  obtain 
the  utmost  accuracy,  to  know  the  weight  of  the  air  which  the  vessel 
contains  ;  and  this  is  deduced  from  its  capacity  ;  for  since  bodies  lose 
in  the  air  a  part  of  their  weight  equal  to  that  of  the  fluid  displaced, 
this  weight  forms  a  quantity  to  be  added  to  all  the  results  ;  and  it  is 
evident  that  if  we  neglect  it,  our  results  are  not  strictly  accurate.  For 
the  same  reason  it  is  necessary  to  observe  the  barometer  and  ther- 
mometer during  the  experiment ;  for  the  weight  of  the  volume  of 
water  displaced  will  be  affected  by  the  state  of  the  air  as  indicated 
by  these  instruments.  Care  must  also  be  taken  to  remove  all  the  air 
contained  in  the  interior  of  the  liquids. 


Liquid  Bodies 


Tin     ... 
Zinc 
Chalk 

Carrara  marble 
Compact  gypsum 
Heavy  spar 
Clay    . 

Rock  crystal     . 
Silex    . 

Fluor  spar        *.  '•' 
Pumice  stone 
Free  Stone 
Common  green  glass 
White  glass    •  s  * 
Flint  Glass   .  ;;     « '. 
Saltpetre         '^.v 
Common  salt 
Ammonia          .  -      . 
Pure  alcohol 
Sulphuric  ether       •  \ 

Wax  .  :v-;   ••*••;••* 

Olive  oil 

Oil  of  turpentine    . 

Green  oak  wood 

Beech     . 

Fir       ... 

Cork 


7,264 

6,H62 

2,25 

2,716 

1,87 

4,3 

1'8     . 
2,653 

2,58 

2,44 

0,914 

2,11 

2,5 

2,4 

3,329 

1,900 

1,918 

1,420 

0,791 

0,716 

0.954 

0,913 

0,792 

0,93 

1,67 

0,85 

0,55 

0,24 


Bergman. 

- 

to  2,32. 

Brisson. 

to  2,29. 

to  4,4. 

to  2,0. 

Brisson. 

to  2,67. 

to  2,60. 

Brisson, 

to  2,56. 

to  2,6. 

to  2,5. 

Brisson. 

Musschenbroek. 


Lowitz. 
to  0,745. 
to  0,960. 
Musschenbroek. 


222.  One  remarkable  circumstance  in  the  chemical  combination 
of  two  bodies  is  that  the  specific  gravity  of  the  combination  cannot 
any  more  than  the  specific  heat  be  determined  a  priori,  because  the 
combination  always  has  a  different  density  from  that  of  the  constitu- 
ent elements.  .If,  for  example,  we  mix  equal  volumes  of  distilled 
water  and  alcohol  of  the  specific  gravity  of  0,824,  we  might  suppose 
that  the  specific  gravity  of  the  mixture  would  be  the  mean  between 
1,000  and  0,824,  or  0,912  ;  but  if  we  make  the  experiment  we  find 
it  to  be  from  0,930  to  0,940 ;  so  that  the  liquid  is  more  dense  after 
the  combination,  and  occupies  a  smaller  space  than  the  constituent 
elements  did. 


Specific  Gravity  of  Solids  and  Liquids.  93 

223.  As  heat  dilates  all  bodies  and  consequently  diminishes  their 
specific  gravity,  the  experiments  should  be  made  at  a  determinate 
temperature  where  accuracy  is  required.     We  usually  take  the  tem- 
perature of  60°  of  Fahrenheit,  because  both  in  summer  and  winter 
this  is  the  most  common  temperature  of  rooms  which  are  occupied, 
and  a  difference  of  one  or  two  degrees  is  not  very  important.* 

224.  To  find  the  capacity  of  a  vessel  or  of  any  other  body.     The 
exact  determination  of  the  weight  of  water,  and  of  the  specific  gravity 
of  bodies,  has  among  other  advantages,  that  of  furnishing  an  easy 
method  of  ascertaining  by  weight  the  capacity  of  all  bodies  with  much 
greater  accuracy  than  by  geometrical  measurement.     If  we  wish  to 
find  the  capacity  of  a  vessel,  we  fill  it  with  water  and  find  the  weight 
of  the  water  it  contains.    This  weight  expressed  in  grains  and  divid- 
ed by  252,525,  gives  the  capacity  of  the  vessel  in  inches.  If  we  aim 
at  great  accuracy  it  is  necessary  to  reduce  the  weight  to  the  tempe- 
rature of  39°,  according  to  the  known  dilatation  of  water. 

225.  If  we  multiply  the  specific  gravity  of  a  body  by  252,525,  we 
find  how  much  a  cubic  inch  of  the  substance  of  this  body  weighs  in 
grains ;  and  if  we  know  the  absolute  weight  of  the  body,  we  determine 
its  capacity  in  cubic  inches  by  dividing  the  whole  weight  by  the 
weight  of  a  cubic  inch.     Such  is  the  general  utility  of  this  method  of 
determining  specific  gravities.f 


*  It  would  sometimes  be  very  difficult  to  obtain  artificially  this 
mean  temperature  of  60°,  and  it  would  be  attended  with  much  trou- 
ble to  preserve  it  ;  but  by  a  short  calculation  we  may  avoid  this  in- 
convenience ;  for  if  we  know  the  dilatations  of  the  bodies  on  which 
we  operate,  which  is  indispensable,  it  is  easy,  by  observing  the  tem- 
perature at  which  the  experiment  is  made,  to  reduce  all  the  weights 
to  the  temperature  required  and  thus  find  their  ratios. 

t  Let  V  be  the  material  capacity  or  volume  of  a  body,  S  its  spe- 
cific gravity,  p  its  absolute  weight  in  grains,  and  252,525  S  the  weight 
of  a  cubic  inch  of  the  body.  Then 


V- 

~ 


252,525  S* 

If  two  of  the  three  quantities   V,  p,  S,  are  given,  we  easily  find  the 
third  ;  for 

p  5=  252,525  SV>  and  S  = 


94  Liquid  Bodies. 


CHAPTER  XXIII. 

Equilibrium  of  Liquids,  or  First  Principles  of  Hydrostatics. 

226.  THE  essential  mechanical  character  of  a  liquid  is  the  perfect 
freedom  with  which  the  parts  move  among  themselves.  Hence  we 
deduce  the  following  principle,  which  may  be  considered  as  the 
foundation  of  the  theory  of  the  equilibrium  and  motion  of  liquids. 
Every  pressure  which  is  exerted  upon  a  liquid,  not  only  acts  in  its 
proper  direction,  but  propagates  itself  uniformly  in  all  directions  to 
every  part  of  the  liquid.* 

In  a  heavy  liquid  each  particle  situated  below  the  surface,  for  ex- 
ample, in  A  (jig.  30)  is  pressed  by  the  weight  of  the  column  of 
liquid  AB  which  is  above  it.  This  particle  presses  with  the  same 
force  and  in  all  directions,  the  rest  of  the  liquid  with  which  it  is  sur- 
rounded. 

If  we  cause  a  horizontal  plane  CD  to  pass  through  .#,  every  other 
particle  E,  which  is  situated  in  this  plane,  must  be  pressed  with  the 
same  force,  if  no  motion  takes  place.  Hence  we  deduce  the  princi- 
ple theorem  of  Hydrostatics,  that  the  surface  of  a  heavy  liquid  must 
be  horizontal,  in  order  that  the  liquid  may  be  in  equilibrium. 

This  theorem  supposes  the  directions  of  the  forces  to  be  perfectly 
parallel.  If  we  suppose  a  celestial  body  formed  solely  of  water  and 
in  a  state  of  rest,  its  surface  must  be  spherical ;  but  on  the  contrary, 
if  it  moves  about  its  axis,  it  will  take  an  oblate  form  in  consequence 
of  the  centrifugal  force  which  tends  to  make  its  particles  separate 

*  This  principle  of  the  equality  of  pressure  in  all  directions  may 
be  presented  in  a  manner  still  more  simple,  from  the  consideration  of 
equilibrium.  The  most  certain  fact  with  which  we  are  acquainted  as 
to  the  constitution  of  liquids  is  their  extreme  mobility.  If,  therefore, 
their  particles  are  in  equilibrium,  each  one  must  be  equally  pressed 
on  all  sides ;  if  not,  since  it  is  perfectly  moveable,  it  would  yield  to 
the  superior  pressure.  It  is  understood  that  in  estimating  the  forces 
which  act  upon  a  liquid,  we  comprehend  the  impenetrability  of  the 
particles,  which  causes  them  to  resist  one  another  ;  and  also  the  im- 
penetrability of  the  sides  of  the  vessel  which  support  the  same  pres- 
sure as  the  contiguous  particles  of  water. 


First  Principles  of  Hydrostatics.  95 

from  the  body  at  the  equator  j  and  this  form  will  be  the  more  flat- 
tened according  as  it  turns  with  greater  velocity.  See  articles  119, 
120. 

227.  The  theorem  is  true,  whatever  be  the  form  of  the  vessel, 
and  in  whatever  manner  its  interior  is  divided,  provided  only  that  the 
portions  of  the  liquids  in  the  different  compartments  have  a  communi- 
cation with  each  other.  If,  for  example,  the  line  EF  represent  a  thin 
partition,  separating  the  liquid  through  the  whole  extent  of  the  vessel ; 
according  to  the  third  law  of  Newton,  this  surface  will  resist  just  as 
much  as  it  is  pressed  ;  that  is,  it  will  act  precisely  as  the  particles  of 
the  liquid  would  do  if  they  were  in  its  place.  Such  a  partition,  there- 
fore, will  not  destroy  the  equilibrium.  We  may  accordingly  introduce 
into  the  vessel  as  many  partitions  of  this  kind  as  we  please,  and  it  is 
obvious  that  in  all  the  compartments  which  communicate  with  each 
other,  the  liquid  will  always  rise  to  the  same  height.     Consequently 
liquids  must  always  rise  to  the  same  height  in  recurved  tubes,  what- 
ever be  their  form,  curvature,  and  dimensions.* 

228.  The  banks  of  rivers  are  rarely  formed  of  substances  impen- 
etrable to  water.     It  is  on  this  account  that  we  always  find  subter- 
ranean water  in  their  neighbourhood.     It  is  evident  from  the  preced- 
ing theorem  that  this  water  must  have  the  same  height  with  the  water 
in  the  river,  though  a  sudden  increase  or  diminution  of  the  latter  may 
produce  a  temporary  difference.     The  subterranean  water  does  not 
proceed  wholly  from  the  river,  but  also  from  rain  and  snow ;  and  con- 
sequently, according  to  circumstances,  it  may  furnish  water  to  the 
river  or  take  water  from  it.     Local  circumstances  determine  to  what 
distance  and  to  what  depth  this  influence  may  extend. 

It  is  a  singular  fact  that  the  existence  of  subterranean  waters,  which 
is  a  thing  so  well  known  to  practical  engineers,  is  not  mentioned  in 
any  work  on  physical  science  with  which  I  am  acquainted.  Still  it 
affords  the  most  simple  and  natural  explanation  of  the  production 
and  support  of  fountains  and  rivers,  the  explanation  of  which  has 
often  been  attempted  upon  the  wildest  hypotheses. 

*  There  is  one  exception  to  be  made  for  the  case  in  which  the 
tubes  are  very  narrow  or  capillary,  for  there  the  fluids  do  not  take  an 
exact  level.  But  this  is  owing  to  the  action  of  an  attractive  force 
peculiar  to  the  material  particles  which  compose  the  tube  and  the 
fluid  ;  and  for  the  present  this  force  is  left  out  of  consideration.  It 
will  be  explained  in  another  place. 


96  Liquid  Bodies. 


Pressure  of  a  Liquid  against  the  Bottom  and  Sides  of  a  Vessel 

229.  Since  the  intensity  of  the  pressure  which  each  point  of  a 
liquid  supports  and  exerts  is  determined  by  article  226,  we  may  also 
determine  without  difficulty  the  pressure  which  any  given  surface 
pressed  by  a  liquid,  exerts  and  supports. 

When  the  surface  is  horizontal  it  supports  precisely  the  weight  of 
a  column  of  the  liquid  which  has  for  its  base  the  surface  pressed,  and 
for  its  altitude  the  height  of  the  water  above  the  surface.  If,  for  ex- 
ample, in  the  four  vessels  A,  B,  E,  F,  (Jigs.  31,  32,  33,  34,)  the 
bottom  JIB  is  of  the  same  magnitude,  and  the  liquid  surface  EF  of 
the  same  height  above  the  bottom,  this  bottom  will  support  the  same 
pressure  in  the  vessels ;  and  the  force  of  this  pressure  is  determined 
by  the  weight  of  a  column  of  fluid,  JIB  CD  raised  vertically  above 
the  bottom.  If  we  know  the  extent  of  the  base  JIB,  and  the  altitude 
J1C,  we  easily  find  the  space  occupied  by  the  column  ;  and  if  the 
weight  of  a  cubic  inch  or  a  cubic  foot  of  this  liquid  is  known,  we 
know  at  the  same  time  the  weight  of  the  column.  Figure  34  repre- 
sents an  instrument  by  means  of  which  the  force  of  pressure  is  ren- 
dered sensible. 

230.  The  parts  of  an  oblique  side  JIB  (Jig.  35)  support  an  un- 
equal pressure  answering  to  the  distance  of  each  point  below  the 
surface  of  the  liquid.     If  this  side  have  the  form  of  a  rectangle,  it  is 
evident  that  the  pressure  which  it  supports  is  equal  to  the  weight  of 
a  prism  of  water,  which  has  for  its  base  half  the  square  of  the  alti- 
tude BF  of  the  water,  and  for  its  altitude  the  breadth  of  the  surface 
pressed.     The  total  pressure  is  the  same  upou  a  vertical  as  upon  an 
oblique  side.* 

231.  When  two  or  more  liquids  which  do  not  mix,   as  mercury, 
oil,  and  water,  are  put  into  the  same  vessel,  they  will  take  their  places 
one  above  the  other  according  to  their  specific  gravities.     But  the 
surfaces  which  separate  them  must  be  horizontal  in  a  state  of  equili- 
brium. 


*  Let  us  suppose  that  having  produced  the  line  CA,  we  draw 
BF  perpendicular  to  its  prolongation.  Then  take  AE  =  BF.  Now  if 
we  take  a  point  G  in  the  side,  and  draw  through  it  the  horizontal  line 
HI,  El  is  the  altitude  of  the  column  of  water  which  presses  upon  G. 


Pressure  of  a  JLiquid  upon  Solid  Bodies  immersed  in  it.      97 

232.  If  we  introduce  into  a  recurved  tube  AB  C,  (Jig.  36)  a  very 
heavy  liquid,  as  mercury,*  for  example,  and  pour  into  one  of  the 
branches  of  the  same  tube  another  lighter  liquid,  water  for  example, 
their  surfaces  will  be  horizontal ;  but  the  surface  C  of  the  lighter 
liquid  will  stand  much  higher  than  the  surface  A  of  the  heavier.  If 
we  draw  through  H  where  the  two  liquids  separate,  the  horizontal 
line  DE,  the  pressure  in  order  that  an  equilibrium  may  take  place, 
must  be  equal  at  D  and  E ;  now  this  can  take  place  only  when  the 
altitudes  of  the  two  columns  which  exert  a  pressure  upon  DE,  are 
in  the  inverse  ratio  of  the  specific  gravities. 


Pressure  of  a  Liquid  upon  Solid  Bodies  immersed  in  it. 

233.  Let  us  suppose  in  a  body  of  tranquil  water  BCD  (Jig.  37) 
a  mass  of  water  A  of  a  form  and  magnitude  taken  at  pleasure,  but 
circumscribed  within  a  space  entirely  geometrical  and  distinct  from 
ihe  rest  of  the  fluid.  It  is  evident  that  the  sum  of  the  pressures  ex- 
erted upon  it  by  the  surrounding  water  must  produce  an  upward 
pressure  just  as  great  as  the  weight  of  the  insulated  mass ;  since 
otherwise  this  mass  would  not  remain  in  equilibrium.  If  now  we 
suppose  this  mass  annihilated,  and  its  place  filled  by  a  solid  body  of 
the  same  form  and  magnitude  every  point  of  its  surface  will  be 
equally  pressed  by  the  surrounding  water,  and  will  exert  a  pressure 
as  great  as  the  water  of  which  it  takes  the  place. 


But  as  the  triangles  BAF,  BGI,  are  similar,  as  well  as  BAE,  BGH, 
we  shall  have  AE  :  GH : :  BF  :  BI,  since  the  ratio  BA  :  BG  is  com- 
mon to  both  triangles.  But  since  in  this  proportion  AE  =.  BF,  GH 
must  be  equal  to  BI.  Consequently,  GH  represents  the  pressure 
which  the  point  G  supports.  The  same  may  be  proved  of  every 
point ;  hence  we  conclude  that  the  triangle  BAE  represents  the 
pressure  upon  the  whole  line  AB.  Now  if  the  side  AB  is  a  rectan- 
gle, each  line  parallel  to  the  section  AB  supports  the  same  pressure. 
Consequently,  the  pressure  upon  the  whole  plane  AB  is  the  weight 
of  a  prism  of  water  which  has  ABE  for  its  base,  and  the  length  of 
the  plane  AB  for  its  altitude.  But  the  triangle  ABE  has  its  base  and 
its  altitude  equal  to  each  other  and  to  the  line  BF.  Thus  its  surface 
is  equal  to  half  the  square  of  the  line  BF. 
Elem.  13 


gg  Liquid  Bodies. 

.Under  these  circumstances  the  body  is  acted  upon  by  two  forces, 
one  of  which  is  exerted  upward,  and  is  equal  to  the  weight  of  the 
water  displaced,  and  the  other  is  the  weight  of  the  body  itself  acting 
in  an  opposite  direction.  Hence  we  deduce  the  following  theorem. 
A  body  immersed  in  a  liquid  loses  just  so  much  of  its  weight  as  is 
equal  to  the  weight  of  the  liquid  displaced. 

234.  If  the  body  A  were  just  as  heavy  as  the'water  displaced, 
it  would,  like  the  mass  of  water  itself,  float  freely  in  the  water.  If 
it  were  heavier  than  the  mass  of  water,  it  would  descend,  not  with 
the  whole  force  of  its  weight,  but  with  that  of  its  excess  over  the 
weight  of  the  water  displaced.  If  it  were  lighter,  it  would  rise  to- 
wards the  surface  with  a  force  equal  to  the  excess  of  the  weight  of 
water  displaced  over  its  own  weight. 


Floating  Bodies. 

235.  In  the  last  case  where  the  body  immersed  is  lighter  than  the 
fluid,  it  rises  till  some  portion  passes  the  surface  of  the  water.     By 
the  effect  of  this  ascent  the  quantity  of  water  displaced  is  diminished, 
and  consequently  the  force  which  raises  it ;  there  must  be  a  time, 
therefore,  when  the  weight  of  the  water  displaced  is  equal  to  the 
weight  of  the  body  ;  then  the  body  is  in  a  condition  to  float  upon  the 
liquid. 

236.  But  experience  proves  that  a  body  cannot  float  in  all  situa- 
tions, although  it  is  immersed  to  a  suitable  depth.     To  understand 
the  cause  of  this,  and  in  general  to  assign  a  reason  for  all  the  phe- 
nomena of  floating  bodies,  two  points  are  to  be  particularly  attended  to. 
1.  The  centre  of  gravity  of  the  body  in  which  we  may  suppose  all 
the  weight  concentrated.   2.  The  centre  of  gravity  of  the  water  dis- 
placed, in  which  we  may  suppose  to  be  concentrated  all  the  force 
which  tends  to  raise  the  body.     The  first  of  these  points  remains 
always  in  the  same  place  in  the  body  ;  but  the  second  changes  its 
situation  according  to  the  changes  which  take  place  in  the  form  and 
situation  of  the  parts  of  the  body  which  are  immersed.     If  these  two 
points  are  not  in  the  same  vertical  line,  the  body  cannot  float  on  the 
surface  of  the  liquid.     If  the  first  point  is  placed  vertically  above  the 
second,  still  the  body  does  not  float  in  a  stable  manner.  It  is  also  ne- 
cessary that  the  circumstances  of  the  body  be  such,  that  if  its  position 
be  changed  by  an  infinitely  small  quantity,  it  will  naturally  return  to  it 


Hydrostatic  Balance  and  Hydrometer.  99 

by  a  series  of  oscillations.  If  the  centre  of  gravity  is  vertically  be- 
low that  of  the  water  displaced,  the  body  must  necessarily  float  and 
in  a  stable  manner.  This  condition  should  be  carefully  attended  to 
in  lading  and  managing  a  vessel.  [On  this  subject  see  Cam.  Mech. 
art.  437,  &c. 


CHAPTER  XXIV. 

Hydrostatic  Balance  and  Hydrometer. 

237.  A  BALANCE  fitted  to  weigh  bodies  under  water,  is  called  a 
hydrostatic  balance.     To  effect  this  object  it  is  only  necessary  to 
fasten  small  hooks  below  a  common  but  accurate  balance.     The 
body  to  be  weighed  is  attached  to  a  very  fine  thread   or  hair,  the 
weight  of  which  is  so  inconsiderable  compared  with  the  entire  mass 
of  the  body  that  it  may  be  neglected.     This  is  suspended  under  one 
of  the  scales  in  such  a  manner,  that  it  can  be  weighed  in  the  air  or 
water  at  pleasure. 

238.  To  find  the  volume  of  a  solid  body.     We  first  weigh  it  in 
air  with  the  hydrostatic  balance  to  which  it  is  attached  by  a  fine 
thread  or  hair.    Then,  without  detaching  it,  we  immerse  it  in  water  ; 
and  as  it  loses  a  part  of  its  weight,  we  add  to  the  scale  below  which 
it  is  suspended,  the  weight  necessary  to  restore  the   equilibrium. 
We  thus  ascertain  how  much  water  the  body  has  displaced ;  and 
this  additional  weight,  expressed  in  grains  and  divided  by  252,525, 
will  give  the  volume  of  the  body  in  cubic  inches. 

239.  To  find  the  specific  gravity  of  water.     When  the  volume  of 
the  body  immersed  is  known,  the  weight  added  indicates  how  much 
water  is  displaced.    This  is  the  method  by  which  the  specific  gravity 
of  water  is  ordinarily  determined. 

240.  To  find  the  specific  gravity  of  a  solid.     We  first  weigh  it 
in  air,  and  then  see  how  much  weight  it  loses  in  water.     The  first 
divided  by  the  last  gives  its  specific  gravity.     But  it  is  taken  for 
granted  that  the  body  is  heavier  than  water,  and  of  such  a  nature 
that  water  neither  dissolves  nor  decomposes  it.     This  method   is 
therefore,  particularly  applicable  when  the  body  is  too  large  to  be 
introduced  into  a  vessel.* 

*  We  might  even  in  this  case  make  use  of  the  method  of  Klaproth 
by  substituting  a  cylindrical  vessel,  closed  air-tight,  instead  of  one 


100  Liquid  Bodies. 

241.  When  a  body  is  mechanically  composed  of  two  known  sub- 
stances to  find  by  means  of  the  hydrostatic  balance,  how  much  it  con- 
tains of  each.  Archimedes,  who  may  be  regarded  as  the  inventor 
of  hydrostatics,  found  that  18  pounds  of  gold  being  weighed  under 
water,  lost  one  pound;  18  pounds  of  silver  lost  1£;  and  a  crown 
weighing  18  pounds,  which  was  composed  of  silver  covered  with  a 
thick  gold  leaf,  lost  1|.  Hence  he  concluded,  by  the  rule  of  fellow- 
ship, that  the  quantity  of  silver  was  to  that  of  the  gold  as  the  differ- 
ences of  the  three  numbers  1,  1£,  H,  that  is,  as  2  to  1  ;  and  that 
consequently  the  crown  was  composed  of  \  gold  and  f  silver.  This 
method  can  be  employed  only  when  the  two  substances  are  mechan- 
ically mixed,  and  not  when  they  are  chemically  combined.  In  the 
latter  case  it  would  give  erroneous  results.* 


of  an  indeterminate  form.  When  this  vessel  is  filled  with  water,  we 
close  it  by  passing  horizontally  over  its  orifice,  a  plate  of  ground 
glass,  which  excludes  all  the  water  that  does  not  make  a  part  of  its 
capacity.  In  this  way  the  cylinder  may  be  very  exactly  closed,  and 
all  the  water  be  removed  which  may  have  adhered  to  its  surface. 

*  Suppose  a  mass  B,  whose  weight  is  p,  composed  of  two  sub- 
stances A,  C ;  let  x  be  the  quantity  of  A  ;  then  p  —  x  will  be  the 
quantity  of  C.  We  find,  by  experiment,  that  the  weight  p}  when  it 
consists  only  of  the  substance  A,  loses  a  in  water  ;  that  the  weight 
p  of  the  compound  body  loses  6,  and  that  the  weight  p  of  the  sub- 
stance C  loses  c.  The  question  is  to  find  x.  We  have  the  propor- 
tion p  :  x  : :  a :  ;  that  is,  if  the  weight  p  of  the  body  A  loses  a, 

the  weight  x  loses  — . 

the  weight  p  —  x  of  the  body  C  loses    -JT- f.      The  compound 

body,  therefore,  loses  in  all  —  -| LEJIL^l  =  b .  whence  we  easily 

obtain  the  value  z  =  -^i-Zlii.     This  formula  leads  precisely  to 

the  rule  given  in  the  text :  for  we  have  »  —  x  =  P  (a       JO .  an(j 

«  —  c    • 
consequently  a  —  b  :b  —  c  :  :p  —  x  :  x. 


Hydrometer.  101 


Areometer  or  Hydrometer. 

242.  A  glass  vessel  JlB,  of  the  form  represented  in  figure  38, 
may  be  sufficiently  light,  not  only  to  float  on  the  water,  but  also  to 
sustain  itself  there,  when  it  has  at  its  lower  extremity  It,  a  weight  of 
lead  or  mercury.     By  means  of  this  weight,  the  centre  of  gravity 
may  be  carried  so  near  the  bottom  as  to  make  the  instrument  float 
with  stability  in  a  vertical  position.     Now  we  have  seen  that  a  float- 
ing body  always  displaces  a  weight  of  liquid  equal  to  its  own  weight. 
It  is  evident,  therefore,  that  such  an  instrument  will  be  immersed  to 
a  greater  depth  in  a  light  liquid  than  in  a  heavy  one ;  hence  it  will 
be  seen  that  the  instrument  may  be  disposed  in  such  a  manner 
as  to  indicate  the  specific  gravity  of  the  liquid  by  the  depth  to  which 
it  sinks.     For  this  purpose  we  introduce  into  the  tube  AC  a  paper 
containing  a  scale  which  indicates  immediately  the  specific  gravity. 
This  instrument  is  called  a  areometer  or  hydrometer. 

243.  The  use  of  the  hydrometer  in  estimating  the  specific  gravity 
of  liquids,  will  appear,  from  what  has  already  been  said,  to  be  super- 
fluous.    But  we  commonly  employ  it  for  a  different  purpose.     For 
example,  in  liquid  mixtures,  such  as  beer,  wine,  brandy,  saline  solu- 
tions, &c.,  the  specific  gravity  changes  with  the  proportion  of  the 
constituent  principles ;  and  it  is  often  very  important  for  scientific, 
economical,  and  mercantile  purposes,  to  know  how  much  such  a 
liquid  contains  of  each  of  its  constituent  principles.     To  make  this 
estimate,  we  commonly  have  recourse  to  the  hydrometer.     But  it  is 
evident  that  there  must  be  a  different  scale  and  arrangement  for  each 
particular  application.     Hence  the  instrument  has  different  names ; 
as  alcoholometer,  vinometer,  &c.     It  is  also  called  assay-instrument, 
gravimeter,  and  pese-liqueur. 

.  244.  The  description  of  one  of  these  instruments  will  suffice  for 
all.  We  shall  take  the  alcoholometer,  and  confine  ourselves  to  a 
general  description  of  its  construction.  Suppose  the  instrument  im- 
mersed first  in  distilled  water  and  then  in  alcohol.  In  the  first  it 
sinks  to  zero  ;  in  the  second  to  100.  We  then  form  mixtures  con- 
taining 10  parts  of  alcohol  and  90  of  water ;  20  of  alcohol  and  80 
of  water,  and  so  on  to  90  of  alcohol  and  10  of  water.  We  immerse 
the  instrument  in  each  of  these  mixtures,  and  having  observed  how 
low  it  sinks,  we  mark  on  the  scale  the  numbers  10,  20,  30,  &tc. 
The  intervals  will  be  unequal ;  but  as  they  only  increase  slowly  we 


102  Liquid  Bodies. 

may  still  divide  each  of  them  into  10  equal  parts,  and  we  shall  thus 
have  an  instrument  which  will  indicate  immediately  how  many  parts 
of  alcohol  are  contained  in  a  mixture  of  water  and  alcohol.  In  order 
to  render  the  degrees  larger  and  more  conspicuous  two  instruments 
are  often  employed,  one  ranging  from  0  to  50°,  and  the  other  from 
50°  to  100°. 

245.  What  we  have  said  will  convey  a  general  idea  of  instruments 
of  this  kind.     They  serve  to  point  out  the  proportion  of  one  of  the 
constituent  principles,  as  salt,  acid,  &c.    As  to  hydrometers  for  wine 
and  spirit,  they  only  represent  arbitrary  degrees  of  goodness.    Even 
the  areometers  of  Baume  indicate  nothing  more  ;  since  their  scales 
have  equal  parts,  and  only  the  two  extreme  points  are  determined 
with  precision  by  weighing ;  whence  these  areometers  will  at  least 
agree  among  themselves. 

246.  There  is  still  another  kind  of  areometer  without  a  scale,  call- 
ed Fahrenheit's.     It  differs  from  the  preceding  in  having  only  one 
mark,  which  indicates  the  depth  to  which  the  instrument  sinks  in 
the  lightest  liquid,  and  in  having  above  the  tube  a  small  trencher  to 
receive  the  weights.     In  fluids  where  it  does  not  sink  to  the  above 
mark,  we  force  it  to  take  this  situation  by  means  of  weights  placed 
upon  the  trencher.     This  simple  apparatus  furnishes  a  very  conven- 
ient method  of  comparing  the  specific  gravities  of  liquids.     We  first 
weigh  the  instrument  itself.     Suppose  it  weighs  460  grains.     Then 
we  immerse  it  in  distilled  water,  and  add  weights  till  it  sinks  to  the 
mark.     Suppose  it  requires  1 04  grains.     Then  we  know  that  the 
instrument  displaces  460  -\-  104  or  564  grains  of  water.     If  we 
find  it  necessary  to  add  160  grains  for  another  liquid,  we  know  that 
the  instrument  displaces  460  -+-  160  or  620  grains  of  this  liquid. 
620  grains,  therefore,  of  this  liquid  fill  the  same  space  as  564  of  the 
other.     Consequently  its  specific  gravity  is  |f  £  =  1,099. 

Nicholson  has  lately  made  an  ingenious  change  in  this  areometer, 
and  thus  rendered  it  a  convenient  instrument  for  estimating  exactly 
the  specific  gravity  of  solid  bodies. 

247.  In  using  all  these  instruments  great  attention  must  be  paid 
to  the  temperature,  as  has  been  before  observed. 


Capillary  Attraction.  103 

CHAPTER  XXV. 

Influence  of  Adhesion  and  Cohesion  upon  Hydrostatic  Phenomena. 

248.  IF  we  suspend  plates  of  glass,  marble,  or  metal,  horizontally 
from  a  hydrostatic  balance  ;  and  having  balanced  them  by  means  of 
weights,  if  we  cause  the  plates  to  touch  the  surface  of  a  liquid,  we 
find  that  it  requires  new  weights  to  make  them  separate  from  this 
liquid.     The  solid  body,  therefore,  attaches  itself  to  the  surface  of 
the  liquid,  which  is  undoubtedly  the  effect  of  an  affinity  exerted  be- 
tween them.     But  it  follows  also  from  this  experiment,  that  the 
particles  of  the  liquid  adhere  together  with  a  certain  force,  since 
otherwise  the  solid  body  would  always  take  away   a  part  of  the 
liquid  with  it,  and  since  in  order  to  effect  the  separation,  it  would  be 
necessary  to  add  just  as  much  weight  as  that  of  the  liquid  separated. 
But  the  result  of  the  experiment  is  entirely  different.    Glass,  marble, 
and  wood,  actually  take  away  a  portion  of  water,  of  alcohol,  and  of 
most  liquids  with  which  they  are  put  in  contact ;  that  is,  they  are 
moistened,  but  the  weight  of  the  liquid  removed,  is  much  less  than 
that  employed  to  effect  the  separation.     Mercury  does  not  even  wet 
these  bodies,  and  yet  it  requires  a  considerable  weight  to  detach 
them  from  its  surface. 

249.  From   the  universality  of  this  phenomenon,  we  conclude 
that  there  exists  a  reciprocal  attraction  or  affinity  between  the  par- 
ticles of  all  solid  bodies  and  liquids.     In  like  manner  the  property 
which  the  particles  of  each  liquid  possesses,  of  adhering  together 
with  a  certain  force,  is  the  consequence  of  an  interior  cohesion,  or 
simply  of  an  exterior  pressure.  We  shah1  call  this  phenomenon  attrac- 
tion, but  without  intending  to  designate  by  the  word  any  tiling  more 
than  the  fact  itself. 

*250.  In  what  we  have  hitherto  said  respecting  the  general  condi- 
tions of  the  equilibrium  of  liquids,  we  have  had  regard  to  gravity  alone. 
But  the  attractive  forces  of  which  we  have  spoken,  introduce  modi- 
fications into  these  phenomena,  which  we  are  now  to  consider.  As 
they  are  very  various  in  their  details,  although  they  all  depend  upon 
the  same  general  cause,  philosophers  have  sought  to  explain  them  in 
many  different  ways.  But  Laplace  is  the  first  who  made  known  the 
real  cause,  and  submitted  the  whole  to  rigorous  calculation.  Not 
having  room  to  trace  here  the  whole  course  of  this  profound  analysis, 

*  The  remainder  of  this  chapter  was  added  by  Biot. 


104  Liquid  Bodies. 

we  shall  endeavour  to  lay  down  the  fundamental  principles  and 
state  the  most  important  results. 

251.  If  in  a  mass  of  tranquil  water  the  surface  of  which  is  horizon- 
tal, we  immerse  vertically  a  tube  of  glass  of  a  very  small  bore  called 
capillary,  the  water  immediately  rises  in  the  interior  of  the  tube,  and 
supports  itself  there  above  its  proper  level.    This  elevation  is  greater 
in  proportion  as  the  diameter  of  the  tube  is  smaller,  and  follows  ex- 
actly the  inverse  ratio  of  this  diameter.     This  is  the  result  of  experi- 
ment, and  it  is  the  most  simple  effect  of  capillary  attraction.     We 
cannot  suppose  that  this  phenomenon  is  owing  to  die  action  of  the 
air,  for  the  same  takes  place  under  the  receiver  of  an  air-pump.  We 
are  obliged,  therefore,  to  regard  it  as  the  result  of  the  attractive  forces 
either  of  the  water  or  of  the  glass,  or  of  both  these  bodies  ;  and  such 
was  also  the  idea  of  Newton.     But  this  great  man  did  not  state  pre- 
cisely in  what  this  attraction  consisted,  nor  how  it  operated.     It  is 
even  evident  from  what  he  has  said  of  the  ascent  of  water  between 
glass  plates,  and  of  the  motion  of  a  drop  of  orange  oil  between  two 
planes  slightly  inclined  to  each  other,  that  he  did  not  know  the  true 
cause  of  these  effects.     Clairaut  is  the  only  geometer  who  has  since 
occupied  himself  with  this  problem.     In  his  fine  work  on  The  Fig- 
ure of  the  Earth,  he  treated  it  as  a  true  question  of  hydrostatics,  and 
analyzed  in  a  very  exact  manner  the  different  forces  of  attraction  01 
gravity,  which  combine  to  determine  the  ascent  of  the  liquid.     Bui 
it  seems  that  his  ingenious  mind  was  led  astray  by  the  false  idea  thai 
the  attractive  action  of  the  tube  might  extend  even  to  the  centre  ol 
the  liquid  column  raised  by  capillary  attraction.    Now  this  is  not  in 
fact  the  case  ;  for  the  liquid  always  mounts  to  the  same  height  in 
a  tube  of  the  same  substance  and  the  same  diameter,  whether  we 
choose  a  thick  or  a  thin  one  ;  so  that  the  strata  of  glass  which  are 
at  a  sensible  distance  from  the  interior  surface  produce  no  apprecia- 
ble affect.     This  fact  which  is  abundantly  verified,  shows,  therefore, 
that  the  attractive  force  of  the  glass,  or  generally  of  the  substance  oi 
the  tube,  decreases  very  rapidly,  as  the  distance  increases  ;  so  that 
its  effect  is  sensible  only  very  near  the  point  of  contact,  and  becomes 
nothing  upon  the  particles  at  an  infinitely  small  distance.     In  this 
respect  the  force  in  question  is  precisely  similar  to  what  the  chemists 
term  affinity.     This  idea,  founded  on  experiment,  is  the  basis  oi 
the  theory  of  Laplace. 

252.  By  admitting  it,  we  see  at  once,  that  the  small  liquid  column, 
which  occupies  the  axis  of  a  capillary  tube  cannot  be  thus  sustained 


Capillary  Attraction.  105 

above  its  level  by  the  attraction  of  the  sides ;  for  this  tube,  though 
capillary,  having  still  a  magnitude  sensible  to  our  eyes,  the  affinity  of 
the  substance  composing  it  cannot  extend  so  far.  We  must  conclude 
then  that  this  column  is  thus  elevated  by  the  action  of  the  water  upon 
itself.  The  question  then  is,  how  can  this  action  produce  such  an 
effect.  The  answer  to  this  question  constitutes  the  discovery  of  La- 
place. In  order  to  understand  it,  let  us  consider  the  manner  in 
which  a  precisely  analogous  action  is  produced,  that  of  bodies  upon 
light.  A  luminous  particle,  when  it  is  at  a  sensible  distance  from 
a  body,  does  not  experience  any  appreciable  action ;  but  when  it 
approaches  to  a  contact,  the  affinity  begins  to  manifest  itself.  The 
particle  becomes  more  and  more  attracted  towards  the  surface  of 
the  body,  by  the  action  of  the  matter  of  which  it  is  composed.  At 
length  enters  and  penetrates  the  interior.  This  action  of  bodies 
upon  light  becomes  very  evident  in  the  phenomenon  called  refrac- 
tion. Setting  out  from  these  principles,  we  determine  with  the 
utmost  accuracy  by  calculation  alone,  the  march  of  the  refracted 
ray.  Now  this  attraction  at  small  distances  is  not  only  exerted  upon 
the  particles  of  light,  but  also  in  the  same  manner,  upon  all  mate- 
rial particles  which  come  in  contact  with  the  surface  of  bodies.  It 
acts,  therefore,  upon  the  particles  which  compose  this  surface. 

253.  Accordingly,  when  a  tranquil  liquid  naturally  takes  a  horizon- 
tal surface,  we  must  suppose  that  this  liquid  exerts  a  particular  action 
upon  itself,  independently  of  terrestrial  gravity.  This  action  tends  to 
make  the  particles  of  the  surface  enter  into  the  interior  of  the  fluid, 
and  would  actually  produce  this  effect  were  it  not  for  the  resistance 
which  results  from  impenetrability.  Now  when  water  rises  in  a  ca- 
pillary tube,  it  does  not  present  a  plane  surface,  but  that  of  a  concave 
meniscus,  nearly  approaching  to  a  hemisphere.  In  this  state  it  still 
exerts  upon  the  particles  ol  its  surface  a  perpendicular  action  from 
without  inward.  But  is  this  action  equal  to  that  which  would  result 
from  a  plane  surface  ?  This  we  must  ascertain  before  we  can  deter- 
mine the  conditions  of  equilibrium,  and  this  accordingly  was  the 
point  first  examined  by  Laplace. 

The  method  employed  is  that  described  in  his  Mecanique  Celeste 
for  calculating  the  attractions  of  spheriods.  He  first  proved  that  a 
body  terminated  by  a  sphere,  or  by  any  portion  of  a  sphere  of  sensi- 
ble extent,  exerts,  from  without  inward,  upon  the  particles  of  its 
surface  an  action  different  from  that  of  a  plane  surface.  This  action 
is  more  feeble  if  the  surface  is  concave,  as  when  water  rises  in 

Elem.  1 4 


i06  Liquid  Bodies. 

glass  tubes ;  and  stronger  when  it  is  convex,  as  when  mercury  i: 
depressed  in  a  tube  which  is  not  perfectly  dry.  The  difference  o 
these  forces  is  the  same  in  both  cases.  It  is  reciprocally  proportiona 
to  the  radius  of  the  sphere,  and  always  very  small  compared  will 
the  action  of  the  plane.  In  order  to  form  an  idea  of  the  cause 
which  produces  it,  we  may  represent  the  column  terminated  by  i 
concave  surface,  as  a  body  terminated  by  a  plane,  plus  a  concave 
meniscus  at  its  upper  extremity  ;  and  the  column  terminated  by  r 
convex  meniscus,  as  a  body  terminated  by  a  plane  minus  a  concave 
meniscus  turned  downward.  Now  the  attraction  of  this  additiona 
meniscus  is  always  the  same,  and  always  tends  to  elevate  the  fluic 
column,  in  whatever  direction  its  concavity  turns.  But  in  the  firsi 
case  it  is  necessary  to  subtract  its  effect  from  that  of  the  plane,  ir 
order  to  have  the  action  of  the  fluid  upon  itself  from  without  in  ware 
and  from  above  downward ;  whereas  in  the  second  case,  we  musi 
add  it  to  the  action  of  the  plane  upon  itself,  since,  not  being  oc- 
cupied by  the  fluid,  there  results  a  diminution  in  the  ascensiona 
force,  and  consequently  an  augmentation  in  the  attractive  force  o: 
the  fluid  for  itself,  the  latter  being  opposed  to  the  former.  If  the 
surface  is  not  spherical,  its  action  upon  itself  is  still  made  up  of  twc 
terms,  one  of  which  represents  the  action  of  the  plane,  and  the  other 
according  as  it  is  negative  or  positive,  that  of  the  concave  or  convex 
meniscus.  This  second  term,  always  very  small  with  respect  to  the 
first,  is  half  the  sum  of  the  actions  of  two  spheres,  having  for  their  radi 
the  greatest  and  least  osculating  radii  of  the  surface  at  the  point  in 
question.  From  this  law,  Laplace  easily  determined  the  partial 
differential  equation,  which  expresses  the  nature  of  the  surface,  and 
by  an  approximate  integration  suited  to  each  circumstance,  he  de- 
duced the  form  of  this  surface,  and  the  action  of  the  fluid  upon 
itself.  It  follows  from  this  analysis,  that  the  term  which  expresses 
the  action  of  the  meniscus  upon  the  fluid  column,  placed  at  the 
centre  of  a  capillary  tube,  is  reciprocally  proportional  to  the  diametei 
of  the  tube. 

254.  Setting  out  from  the  results  furnished  by  the  calculus,  we 
are  able  to  give  a  satisfactory  explanation  of  the  phenomena  of  ca- 
pillary tubes.  Beginning  with  the  cas«  in  which  the  fluid  is  elevated 
above  the  natural  level,  and  which  requires  the  upper  extremity  o! 
the  fluid  column  to  be  concave,  we  suppose  an  infinitely  small  fila- 
ment of  fluid  extending  from  the  lowest  point  of  the  meniscus  along 
the  axis  of  the  tube,  and  then  returning  in  any  manner  through  the 


Capillary  Attraction.  107 

mass  of  the  liquid  to  the  free  surface.  The  fluid  being  in  a  state  of 
equilibrium,  this  filament  will  be  in  a  state  of  equilibrium.  But  it  is 
pressed  downward  at  the  two  extremities  with  unequal  forces.  The 
force  exerted  at  the  free  surface  is  the  action  of  a  body  terminated 
by  a  plane  surface  ;  the  other  in  the  interior  of  the  tube  is  the  action 
of  the  same  body  terminated  by  a  concave  surface,  or  one  in  which 
there  is  a  contrary  attraction  upward,  the  little  annulus  cut  off  by  a 
horizontal  plane  passing  through  the  lowest  point  of  the  meniscus, 
and  which  is  supported  by  the  attraction  of  the  glass,  exerting  an 
upward  force.  It  is  necessary,  therefore,  in  order  that  an  equilibrium 
may  take  place  that  the  fluid  should  rise  in  the  tube  till  the  weight 
of  the  column  thus  elevated  above  the  natural  level,  should  compen- 
sate for  this  difference  in  the  downward  pressures  exerted  at  the  two 
extremities  of  the  filament.  This  difference  is  in  the  inverse  ratio  of 
the  diameter  of  the  tube  ;  the  height  of  the  small  column  must  ac- 
cordingly be  in  the  same  ratio  ;  and  this  is  conformable  to  the  results 
of  our  observation. 

255.  If  the  fluid  surface  were  convex  instead  of  being  concave, 
the  results  would  be  contrary.  In  this  case,  Laplace  has  demon- 
strated that  its  action  would  be  greater  than  that  of  the  plane,  always 
in  the  inverse  ratio  of  the  diameter  of  the  tube.  Consequently,  if 
we  suppose  a  liquid  to  take  this  form  in  a  capillary  tube,  by  repeat- 
ing the  above  reasoning  with  this  simple  modification,  we  shall  see 
that  the  small  curvilinear  filament  is  still  pressed  unequally  at  its  two 
extremities,  more  at  the  convex  surface,  and  less  at  the  horizontal. 
Hence  it  follows  that  in  order  to  an  equilibrium,  the  fluid  must  be 
depressed  in  the  tube  where  the  action  is  strongest,  until  it  produces 
a  difference  of  level,  which  will  compensate  for  the  weakness  of  the 
opposite  force.  The  depression  of  the  fluid  will  therefore  be  as  the 
difference  of  the  two  forces,  that  is,  reciprocally  proportional  to  the 
diameter  of  the  tube ;  and  this  is  what  actually  takes  place,  when 
the  fluid  does  not  wet  the  tube,  and  attach  itself  to  its  sides,  as 
when  we  immerse  a  glass  tube  in  water,  after  having  put  a  thin  coat 
of  oil  over  its  interior  surface  ;  or  when  we  immerse  in  mercury  a 
glass  tube  not  perfectly  dry.  Under  these  circumstances  the  sur- 
face of  the  fluid  within  the  tube  takes  a  convex  form,  and  the  fluid 
is  depressed  below  its  level,  exactly  in  the  inverse  ratio  of  the  diam- 
eter of  the  tube.  But  if  we  remove  the  obstacle  which  prevents  the 
glass  and  the  liquid  from  adhering  to  each  other,  then  the  latter  will 
take  the  concave  form  and  ascend  in  the  tube  above  its  level.  This 


108  Liquid  Bodies. 

happens  even  in  the  case  of  mercury,  when  it  is  well  dried,  and 
when  the  tube  has  been  deprived  of  all  moisture  by  long  boiling. 
Such,  for  example,  are  those  barometric  tubes,  from  which  all  air 
and  vapour  has  been  removed  by  the  repeated  boiling  of  mercury 
in  them.  This  leads  us  to  remark  that  a  single  boiling  is  not  suffi- 
cient for  this  purpose  ;  and  ordinary  barometers  prove  it ;  since  the 
mercury  always  preserves  in  them  a  convex  form. 

256.  The  peculiar  character  of  this  theory  consists  in  this,  that  it 
makes  every  thing  depend  upon  the  form  of  the  surface.    The  nature 
of  the  solid  body  and  that  of  the  fluid  determine  simply  the  direction 
of  the  first  elements,  where  the  fluid  touches  the  solid,  for  it  is  at 
this  point  only  that  their  mutual  attraction  is  sensibly  exerted.  These 
directions  being  given,  they  become  the  same  always  for  the  same 
fluid  and  the  same  solid  substance,  whatever  be  the  figure  of  the 
body  itself  which  is  composed  of  this  substance.     But  beyond  the 
first  elements  and  beyond  the  sphere  of  action  of  the  solid,  the  direc- 
tion of  the  elements  and  the  form  of  the  surface  are  determined 
simply  by  the  action  of  the  fluid  upon  itself. 

All  the  causes,  therefore,  which  by  acting  upon  the  surface  of  the 
glass,  can  change  the  direction  of  the  first  elements,  must  change 
also  the  curvature  of  the  liquid  surface,  and  consequently  the  eleva- 
tion of  the  fluid.  This  explains  the  depression  of  water  in  tubes 
coated  on  the  interior  with  an  oily  substance,  the  elevation  of  mercury 
in  dry  tubes,  and  bs  depression  in  moist  ones.  Friction  may  also 
produce  analogous  effects,  and  Laplace  has  cited  examples  of  this 
kind.  These  effects  are  easily  explained  by  his  theory,  and  instead 
of  being  irregular  and  anomalous,  as  they  appear  at  first,  they  are, 
on  the  contrary,  subjected  to  fixed  laws,  and  may  be  exactly  pre- 
dicted. 

257.  Capillary  phenomena  are  not  confined  to  tubes,  but  take  place 
also  in  planfc  spaces.     Water  rises  and  mercury  falls  between  two 
glass  plates,  placed  at  a  small  distance  from  each  other.     The  law 
of  these  phenomena  is  the  same  as  in  the  case  of  tubes.     The  ele- 
vations and  depressions  are  reciprocally  proportional  to  the  distances 
of  the  plates.     But  there  is  this  singular  difference,  remarked  by 
Newton,  that  the  absolute  effect  is  half  of  what  it  is  in  tubes ;  that  is, 
between  plates  at  the  distance  of  ^V  of  an  inch,  for  example,  the 
water  rises  to  precisely  the  same  height  as  in  a  tube  of  Ty     New- 
ton merely  stated  this  result  in  the  queries  placed  at  the  end  of 
his  Optics  ;  and  although  it  is  very  remarkable,  it  does  not  appear 


Capillary  Attraction.  109 

to  have  arrested  the  attention  of  philosophers  till  Laplace  took  up 
the  subject ;  probably  because  they  confined  themselves  to  the  ca- 
pillary effects  perceived  in  tubes,  without  suspecting  they  had  so 
intimate  a  connexion  with  those  of  plates.  This  singular  relation  is 
easily  deduced  from  the  theory  of  Laplace.  We  have  seen  that  in 
the  case  of  tubes  the  action  of  the  concave  or  convex  surface  upon 
the  column  raised  is  half  the  action  of  two  spheres  which  would  have 
for  their  radii  the  greatest  and  least  osculating  radii  to  the  surface  at 
the  lowest  point.  If  the  tube  is  flattened  in  one  direction,  the  cor- 
responding radius  of  curvature  augments ;  and  finally  becomes  infi- 
nite when  the  tube  is  changed  into  two  parallel  planes  ;  the  part  of 
the  attraction  of  the  surface  which  was  reciprocally  proportional  to 
this  radius,  diappears,  therefore,  by  the  effect  of  this  change,  and 
there  remains  only  the  term  depending  on  the  other  osculating  radius. 
The  attractive  action  is  thus  reduced  one  half.  Such  is  the  simple 
and  rigorous  result  furnished  by  the  theory  of  Laplace. 

258.  This  theory*  explains  also,  with  the  same  simplicity,  all  other 
capillary  phenomena.  Thus,  the  ascension  of  water  in  concentric 
cylinders  or  conical  tubes,  the  curvature  which  it  takes  when  it  ad- 
heres to  a  glass  plane,  the  spherical  form  which  liquid  drops  natur- 
ally take,  the  motion  of  a  drop  between  two  glass  plates  slightly 
inclined,  the  force  which  brings  together  bodies  floating  near  each 
other  on  the  surface  of  water,  the  adhesion  of  plane  discs  to  the  sur- 
face of  liquids,  sometimes  so  strong,  that  it  requires  a  considerable 
force  to  separate  them,  &tc. ;  all  these  various  effects  are  deduced 
from  the  same  formula,  not  in  a  vague  and  conjectural  manner,  but 
with  numerical  exactness ;  and  they  thus  acquire  relations  not  be- 
fore known.  For  example,  we  see  clearly  from  this  theory,  why 
two  parallel  glass  plates,  immersed  in  water  at  a  small  distance  from 
one  another,  tend  to  approach  each  other  even  when  the  water  rises 
between  them.  For  if  we  conceive  between  these  two 'plates,  and 
in  the  axis  of  the  column  raised,  a  small  vertical  filament  recurved 
horizontally  at  its  lower  extremity,  and  terminating  perpendicularly 
to  the  interior  surface  of  one  of  the  plates,  this  filament  will  be 
pressed  differently  at  its  two  extremities.  In  the  first  place  it  will 
be  pressed  horizontally  and  from  without  inward,  by  the  action  re- 
sulting from  the  liquid  in  contact  with  the  plane  surface  of  the  glass. 
Secondly,  at  the  superior  extremity  it  will  be  pressed  from  above 
downward,  by  the  action  of  the  plane  minus  that  of  the  meniscus, 
and  moreover  by  the  weight  of  the  small  column  of  water,  which  is 


110  Liquid  Bodies. 

elevated  in  the  vertical  branch  above  the  point  in  question.  Thus 
if  we  subtract  the  force  of  the  plane  which  presses  the  other  ex- 
tremity, there  still  remains  for  the  pressure  from  without  inward,  the 
action  of  the  meniscus,  minus  the  liquid  column  raised.  These  two 
actions  exactly  compensate  each  other  if  the  point  in  question  is  at 
the  natural  level  of  the  fluid  j  but  equilibrium  does  not  take  place 
above  this  point.  As  we  rise  above  it,  the  distance  from  the  surface 
becoming  smaller,  the  weight  of  the  liquid  column  cannot  compen- 
sate the  attractive  action  of  the  meniscus,  and  the  two  plates,  being 
attracted  towards  the  top  by  this  force,  tend  necessarily  to  approach 
each  other.  Those  who  will  take  the  trouble  to  compare  these 
results  with  the  numerous  explanations  given  by  philosophers,  and 
with  those  of  Newton  himself,  will  perceive  the  advantage  of  a  math- 
ematical theory  over  simple  conjecture. 

259.  Laplace  subjected  his  theory  to  the  most  rigorous  proof  by 
comparing  it  with  experiments.     For  this  purpose,  he  selected  those 
which  were  made  by  Hauksbee,  under  the  inspection  of  Newton,  to 
which  he  also  added  others  still  more  accurate,  made  by  Gay-Lus- 
sac  at  his  request.     Though  the  formulas  were  only  approximate, 
the  agreement  between  them  and  the  results  of  these  formulas  is 
truly  wonderful ;  and  it  is  obvious  that  this  precise  numerical  deter- 
mination of  the  results  is  the  true  touchstone  of  the  theory. 

There  are  no  discoveries  in  the  sciences  which  have  not  sooner 
or  later  some  useful  application.  The  effects  of  capillary  attraction 
are  perceived  in  barometric  tubes ;  and  as  the  surface  of  the  mer- 
cury is  convex,  there  must  result  a  slight  depression  in  the  height  of 
this  column,  which  then  does  not  exactly  indicate  the  weight  of 
the  atmosphere.  This  effect  is  nothing  in  barometers  with  two 
branches,  because  the  two  forces  resulting  from  the  convexity  of  the 
fluid  counterbalance  each  other.  But  it  exists  in  simple  barometers, 
and  may  become  appreciable  in  exact  researches.  Laplace  has  in- 
dicated a  very  easy  process  for  determining  by  experiment  the  cor- 
rections to  be  made  on  this  account  at  all  observed  heights  ;  and  he 
has  moreover  calculated  a  table  in  which  the  value  of  these  correc- 
tions is  expressed  numerically  according  to  the  diameter  of  the  tube. 

260.  It  is  obvious  also  from  the  preceding  remarks,  that  the  heights 
must  be  reckoned  from  the  summit  of  the  convexity  of  the  mercury, 
and  not,  as  some  observers  do,  from  the  point  where  this  convexity 
begins.     Proceeding  according  to  this  second  method,  the  observed 
heights  of  the  mercury  are  all  too  small  by  a  quantity  equal  to  the 


Hydraulics.  Ill 

radius  of  the  meniscus,  which  being  augmented  proportionally  to  the 
difference  of  the  weight  of  the  mercury  and  air,  may  produce  con- 
siderable errors  in  estimating  the  elevations  of  objects. 


CHAPTER  XXVI. 

Motions  of  Liquids,  or  First  Principles  of  Hydraulics. 

261.  WATER  is  subjected  to  many  different  motions,  the  con 
sideration  of  which  is  of  great  interest  to  reflecting  men,  because 
these  effects  have  an  important  influence  upon  the  wants  of  so- 
cial life.     These  motions  are  either  natural  or  artificial.     Springs, 
brooks,  torrents,  rain,  all  the  agitations  of  the  sea,  especially  the 
tides,  as  well  as  constant  and  variable  currents,  are  examples  of  the 
first  kind.     Among  the  artificial  motions,  we  distinguish  particularly 
those  of  water  in  canals  and  in  those  ingenious  hydraulic  machines, 
which  are  found  to  be  of  such  great  utility.  It  belongs  to  mechanical 
philosophy  to  establish  and  confirm  the  principles  of  these  different 
motions.     But  with  respect  to   their  application,  that   part  which 
relates  to  the  natural  motions  belongs  to  physical  geography,  and  the 
other  to  the  science  of  machines. 

262.  Detached  masses  of  liquid  observe  exactly  the  same  laws  as 
solid  bodies,  when  all  their  parts  move  with  an  equal  velocity  and  in 
the  same  direction.     Thus  the  motion  of  a  drop  of  water  which  falls 
with  the  conditions  which  we  have  assigned,  is  absolutely  the  same 
as  that  of  a  solid  mass  under  similar  circumstances.     But  the  essen- 
tial mobility  of  all  the  particles  of  a  liquid,  with  respect   to  one 
another,  renders  it  almost  impossible  for  them  to  have  motions  direct- 
ed the  same  way  and  with  the  same  velocity.     Interior  motions  take 
place,  which  it  is  difficult  to  observe,  and  still  more  difficult  to  calcu- 
late.    These   interior   motions   embarrass   the  theory.     Hydraulic 
experiments  have  also  in  themselves  a  peculiar  difficulty,  arising 
from  the  fact  that  we  cannot  withdraw  the  motions  of  liquids  from 
the  influence  of  all  foreign  forces,  so  easily  as  we  can  those  of  solids ; 
and  that  we  cannot  without  great  trouble,  determine  exactly  by  the 
calculus  what  must  be  the  effect  of  each  of  these  forces. 

263.  The  principal  problem  to  be  solved  in  hydraulics,  relates  to 
the  velocity  with  which  a  liquid  passes  out  through  an  opening  made 


1 12  Liquid  Bodies. 

in  the  bottom  or  sides  of  a  vessel.  Let  ABCV  (fig.  41)  and 
EFGH  (Jig.  42),  be  two  vessels  of  different  heights  AC,  EG, 
which  we  suppose  to  be  filled  with  a  certain  liquid  and  to  be  kept 
full  by  a  constant  influx.  In  the  bottom  CD  and  GH  of  both,  are 
apertures  IK,  LM,  of  the  same  dimensions,  but  very  small  compar- 
ed with  the  extent  of  the  vessels. 

If  then  we  suppose  the  liquid  to  be  acted  upon  by  gravity  alone, 
we  can  very  easily  determine  by  the  general  laws  of  motion,  the 
ratios  of  the  velocities  of  the  masses  of  water  which  flow  through  the 
two  apertures.  For  supposing,  as  we  have  done,  that  the  height  of 
the  liquid  remains  invariable  in  the  two  vessels,  it  is  evident  that  the 
velocities  in  each  will  be  uniform.  The  quantities  which  run  out  in 
equal  times,  are  then  as  the  velocities,  whatever  these  times  may  be. 
Since  in  general,  the  quantity  of  each  motion  is  measured  by  the 
product  of  the  mass  into  the  velocity,  and  since  here  the  masses  are 
proportional  to  the  velocities,  it  is  evident  that  the  quantity  of  motion 
produced  in  any  given  time,  is  as  the  square  of  the  velocity.  But 
the  ratio  of  the  quantities  of  motion  is  also  the  ratio  of  the  moving 
forces.  In  the  cases  we  are  examining,  these  moving  forces  are  the 
weight  of  the  two  columns  of  liquid  which  are  situated  vertically 
above  the  apertures.  Since,  their  bases  are  the  same,  these  columns 
are  as  their  altitudes  AC  and  EG.  The  square  of  the  velocity  in 
IK  must  therefore  be  to  the  square  of  the  velocity  in  LJ\1,  as  A  C  is 
to  EG  ;  that  is,  the  velocities  are  as  the  square  roots  of  the  heights  of 
pressure. 

This  is  the  most  important  principle  of  hydraulics. 

264.  It  may  also  be  demonstrated  by  the  laws  of  accelerated  mo- 
tion, that  the  absolute  velocity  of  a  liquid  flouring  out  by  the  mere 
force  of  gravity,  is  equal  to  the  velocity  which  a  heavy  body  would 
acquire  by  falling  from  the  superior  surface  of  the  liquid  down  to 
the  aperture.* 


*  In  order  to  demonstrate  the  truth  of  this  law,  we  observe  that 
the  total  velocity  of  the  liquid  flowing  out,  as  well  as  all  other  velo- 
cities resulting  from  pressure,  are  not  produced  instantaneously,  but 
observe  an  acceleration  beginning  from  zero.  This  acceleration  is 
uniform  in  the  present  case,  since  we  have  supposed  the  height  of 
pressure  to  remain  the  same.  Our  problem  is,  therefore,  to  be 
solved  by  the  laws  of  uniformly  accelerated  motion.  Now  let  PQIK, 
(Jig.  41)  denote  the  pressing  column;  NOIK  a  small  part  of  this 


Hydraulics.  113 

265.  A-change  in  the  magnitude  of  the  aperture  cannot  change 
this  velocity ;  for  if  we  double  the  aperture,  the  weight  of  the  press- 
ing column  will  also  be  doubled,  it  is  true,  but  at  the  same  time,  the 
mass  to  be  moved  will  be  doubled  in  like  manner. 

It  follows  from  this  that  the  ratio  of  the  magnitude  of  the  aperture 
to  the  extent  of  the  vessel,  has  no  immediate  influence  upon  this 
velocity.  For  if  the  aperture  were  equal  to  the  bottom  of  the  vessel, 
the  inferior  stratum  CD  would  fall  at  the  instant  the  opening  was 
made,  with  the  acceleration  determined  in  the  preceding  article ; 
but  if  the  vessel  were  to  remain  full,  the  velocity  of  the  water  flow- 
ing in  would  be  a  new  force,  to  which  regard  is  had  in  the  funda- 
mental principle.  It  is  for  this  reason  that  we  have  supposed  the 
opening  extremely  small  compared  with  the  extent  of  the  vessel,  in 
order  to  diminish  the  effect  of  this  foreign  force. 

266.  These  laws  do  not  depend  at  all  upon  the  specific  gravity  of 
the  fluid.     If  one  vessel  contain  mercury  and  the  other  water,  both 
being  at  the  same  height,  the  pressure  of  the  mercury  for  equal 
openings  will  be,  it  is  true,  14  times  greater  ;  but  the  mass  being  as 
many  times  more  difficult  to  be  moved,  the  velocity  will  not  be 
altered. 

267.  If  the  vessel  is  not  pierced  at  the  bottom,  but  on  the  side,  as 
at  EF  (Jig.  43),  the  particles  of  water  do  not  flow  out  with  an  equal 
acceleration  through  all  the  points  of  the  opening.     Yet  if  the  open- 


column  taken  at  pleasure.  If  the  mass  NOIK  falls  by  its  own  weight, 
it  must  have,  after  describing  the  path  NI}  a  velocity  c  =.  \/4g  NL 
But  here  the  velocity  which  we  shall  call  x,  must  be  greater,  since  its 
acceleration  is  produced  by  the  weight  of  the  whole  column  PQIK. 
The  acceleration  of  free  descent,  therefore,  the  measure  of  which  is 
g}  must  be  to  the  acceleration  in  the  present  case,  as  the  weight  of 
NOIK  is  to  the  weight  of  PQIK.  The  acceleration  sought  is,  then, 

°"  X  P I 

the  fourth  proportional  to  NI,  PI,  and  g  ;  that  is,  P    „ — ;  so  that  to 

find  a;,  we  have  only  to  substitute  this  value  in  the  place  of  g  in  the 
above  formula,  and  we  have 


*j 


S«-Vi77^ 


It  will  of  course  be  seen  that  this  velocity  is  the  same  as  that  of  a 
heavy  body  falling  freely,  after  haviner  described  the  space^  PI  or 
AC. 

Ekm.  15 


114  Liquid  Bodies. 

ing  is  small,  and  the  point  G  be  in  the  middle  of  it,  we  may,  with 
out  material  error  consider  the  mean  velocity  of  the  liquid  as  tha 
which  belongs  to  the  altitude  BG. 

268.  If  the  opening  is  made  in  the  upper  surface  of  a  horizonta 
projection,  as  GH  (fig.  44)  the  water  spouts  out  with  a  primitiv< 
velocity  perfectly  answering  to  the  principles  which  we  have  estab 
listed. 


Hydraulic  Experiments  confirming  the  preceding  Theory. 

269.  For  these  experiments  we  commonly  make  use  of  prismatii 
or  cylindrical  vessels  ;  the  greater  they  are,  the  better.   The  experi 
ments  are  most  frequently  made  with  water.    The  bottoms  and  side 
of  the  vessels  have  openings  of  different  forms  and  magnitudes  ;  am 
we  also  employ  cylindrical  and  conical  tubes  of  all  dimensions  suitei 
to  the  apertures.     The  vessels  are  always  kept  full  during  the  ex 
periment,  by  a  constant  flowing  in  of  water ;  or  else  the  opening  i 
so  small  compared  with  the  dimensions  of  the  vessel,  that  the  vvate 
may  flow  out  for  several  seconds  without  perceptibly  lowering  th 
water. 

270.  With  such  an  apparatus  we  can  determine  by  experimer 
the  velocity  with  which  the  water  flows  in  each  case.     We  sufFe 
the  water  to  flow  during  10  seconds,  for  example.     The  weight  c 
this  water,  expressed  in  grains,  being  divided  by  252,525,  the  weigl 
of  a  cubic  inch,  will  give  the  number  of  cubic  inches  in  the  mass 
and  this  being  again  divided  by  10,  the  number  of  seconds,  will  giv 
the  solidity  of  the  mass  which  runs  out  in  a  second.     This  mas 
forms  a  column  the  base  of  which  is  the  aperture,  and  the  altitud 
the  space  described  in  a  second  or  the  velocity.     If  then  we  divid 
this  column  by  the  superficial  dimensions  of  the  aperture,  we  hav 
the  velocity  with  which  the  water  flows. 


Influence  of  Forces  different  from  Gravity  upon  Hydraulic  Motion. 

271.  The  theory  here  presented,  rests  upon  principles  so  incor 
testible,  and  the  proofs  of  it  are  so  simple,  that  we  cannot  doubt  ii 
correctness.  Yet  if  we  compare  the  results  of  this  theory  with  es 
periment,  we  do  not  find  them  to  be  completely  verified.  The  fin 


Hydraulics.  115 

principle  of  article  262  is  well  confirmed  by  fact,  since  the  velocities 
of  water  flowing  from  different  heights,  are  in  reality  as  the  square 
roots  of  the  altitudes  of  pressure,  provided  the  apertures  are  of  equal 
dimensions.  But  what  relates  to  the  absolute  velocity  is  never  con- 
formable to  the  law  expressed  in  article  264.  In  most  cases  this 
velocity  is  less,  which  may  be  easily  accounted  for  by  the  obstacles 
which  oppose  it.  But  there  are  also  cases  in  which  it  is  greater ; 
indeed  this  augmentation  is  sometimes  more  than  one  half.  More- 
over, with  the  same  height  of  pressure,  we  find  a  change  of  velocity 
in  every  case  when  we  give  the  aperture  a  different  disposition ; 
when,  for  example,  we  form  it  alternately  by  a  simple  orifice  made 
in  a  thin  plate,  and  by  longer  or  shorter  tubes,  cylindrical  or  conical, 
and,  in  this  last  case,  made  larger  at  the  interior  or  exterior  extremity. 
Hitherto  we  have  not  been  able  to  reduce  this  difference  to  simple 
principles.  Yet  these  experiments  themselves  prove  that  the  devia- 
tions are  not  occasioned  by  gravity,  but  depend  entirely  upon  foreign 
circumstances.  They  do  not,  therefore,  prove  any  thing  contrary  to 
the  theory  proposed  ;  but  only^how  that  we  have  not  yet  been  able 
to  subject  the  influence  of  these  forces  to  mathematical  laws. 

272.  The  forces  and  circumstances  which  modify  the  primitive 
velocity  of  a  liquid,  originally  acted  upon  by  gravity  alone,  may  be 
comprised  in  what  follows. 

(1.)  Flowing  water  has  to  conquer  the  resistence  of  the  air,  which 
diminishes  its  velocity. 

(2.)  The  motions  which  take  place  in  the  interior  of  each  liquid 
when  flowing,  are  an  important  cause  of  modification.  It  is  difficult 
to  observe  these  forces,  and  still  more  difficult  to  subject  them  to 
exact  laws.  When  a  jet  of  water  issues  through  the  opening  EF, 
(fig'  45)  from  the  vessel  JlBCD,  it  is  not  merely  the  vertical  col- 
umn above  EF  which  falls  ;  but  all  the  water  in  the  vessel,  if  it  is 
not  very  large,  has  a  motion  of  descent.  If  this  vessel  be  of  glass, 
and  we  distribute  small  fragments  of  some  light  substance,  as  sealing- 
wax,  for  example,  through  the  water,  this  motion  becomes  visible. 
Towards  the  top  the  whole  liquid  mass  falls  with  sufficient  uniformi- 
ty, if  the  vessel  be  of  equal  magnitude.  Lower  down  the  motion 
does  not  continue  either  uniform  or  rectilinear  ;  but  the  particles  of 
water  take  nearly  the  directions  represented  in  figure  45.  The  water 
flows,  therefore,  from  all  parts  towards  the  aperture,  and  as  its  mo- 
tions are  partly  opposed  to  each  other  they  must  produce  a  consider^ 
able  retardation  in  the  velocity. 


116 


Liquid  Bodies. 


Tne  interior  motions  become  still  more  varied,  and  the  diminution 
of  velocity  still  more  considerable,  if  the  vessel  is  not  of  the  same 
dimensions  throughout,  especially  if  it  is  of  an  irregular  form,  and 
still  more  if  it  consists  of  a  tube  several  times  recurved. 

Particular  attention  should  be  paid  to  the  form  of  the  jet  as  modi- 
fied by  these  interior  motions.  If  the  aperture  is  simply  pierced  in 
a  thin  plate,  the  jet  immediately  below  it  has  the  form  of  a  truncated 
cone  inverted,  as  EFGH  (Jig.  45) ;  but  in  such  a  manner  that  the 
sides  EG,  FH,  are  curved  inward.  The  dimensions  of  this  cone 
are  very  constant  in  the  circumstances  supposed.  The  smallest 
diameter  of  the  jet  GH  is  0,8  of  the  diameter  EF  of  the  aperture  ; 
now  the  surfaces  of  circles  being  proportional  to  the  squares  of  their 
radii,  the  section  of  the  fluid  column  is  0,64,  or  about  two  thirds  that 
of  the  orifice.  Below  GH  the  fluid  column  dilates.  The  distance 
between  GH  and  EFis  only  equal  to  half  the  diameter  of  the  aper- 
ture EF.  This  phenomenon  is  called  the  contraction  of  the  jets. 
The  velocity  of  the  water  increases  very  rapidly  between  EF  and 
GH;  since  in  GHit  must  be  greatqp  by  one  half  than  in  EF;  for, 
in  equal  times,  the  same  quantity  of  water  passes  through  EF  and 
GH ;  and  since  these  two  sections  are  to  each  other  as  3  to  2  ;  the 
velocities  in  each  must  be  in  the  inverse  ratio,  that  is,  as  2  to  3. 
Experiments  prove  that  the  velocity  of  wnter  in  GH  approaches 
nearly  to  the  velocity  which  belongs  to  the  altitude  of  descent  AC. 
It  appears,  therefore,  that  in  the  section  EG  the  effect  of  all  the 
foreign  forces  has  disappeared,  and  that  the  water  has  then  recovered 
the  velocity  which  it  ought  to  have  from  the  effect  of  gravity  alone. 
This  is  a  very  striking  proof  of  the  exactness  of  the  theory  we  have 
stated. 

(3.)  Lastly,  the  greater  or  less  adhesion  which  take  place  be- 
tween the  vessel  and  the  liquid,  and  that  which  always  exists  be- 
tween the  particles  of  the  liquid,  have  a  much  greater  influence  upon 
the  velocity  of  the  issuing  water  than  we  should  at  first  imagine. 

It  is  undoubtedly  to  this  influence  that  we  are  to  ascribe  the  differ- 
ent velocities  we  observe,  according  as  we  give  different  forms  to  the 
orifice.  It  is  evident  that  these  adhesions  are  obstacles  to  the  motion 
in  most  cases ;  and  when  the  aperture  is  very  small,  all  motion  may 
be  prevented  by  them.  Yet  it  appears  that  under  certain  circum- 
stances these  forces  do  not  diminish  the  motion,  but,  on  the  contrary, 
augment  it.  The  most  remarkable  effect  of  this  kind  takes  place 
when  we  apply  to  the  aperture  a  tube  in  the  form  of  an  inverted 


Hydraulics.  117 

cone,  which  has  the  dimensions  of  the  contracted  jet,  and  when  we 
add  below  this  another  conical  tube  which  spreads  insensibly. 

273.  When  water  spouts  upward,  it  meets  a  particular  obstacle 
besides  those  already  described.     Every  drop  rises  with  a  retarded 
motion.     The  velocity  is,  therefore,  less  in  the  higher  parts  of  the 
jet  than  in  the  lower.     Accordingly  the  more  elevated  portion  of  the 
water  exerts  a  pressure  upon  that  which  is  below,  and  retards  its 
motion.     For  this  reason  the  jet  never  attains  the  height  due  to  the 
primitive  velocity  of  the  water.     Moreover  the  water  which  rises  is 
retarded  still  more  by  that  which  falls,  and  sometimes  it  is  crowded 
back  into  the  orifice  from  which  it  issues.     On  this  account  water 
rises  higher  when  it  does  not  issue  in  an  exactly  vertical  line.    As  to 
the  disposition  of  the  aperture,  experience  has  shown  that  the  one 
best  adapted  to  give  the  jet  a  great  elevation,  is  also  the  most  sim- 
ple ;  that  is,  a  small  orifice  pierced  in  a  thin  plate. 

274.  It  is  a  general  law  for  all  cases,  that  when  a  liquid  issues 
from  a  vessel  the  vessel  itself  suffers  a  pressure  in  the  opposite  direc- 
tion.    This  pressure  may  even  give  the  vessel,  if  it  be  sufficiently 
moveable  a  motion,  in  a  contrary  direction.     This  pressure  still  ex- 
ists when  the  aperture  EF  (jig  43)  is  closed  ;  and  its  intensity  may 
be  estimated  from  what  was  said,  article  230. 

But  whatever  be  its  force,  it  cannot  in  this  last  case  produce  any 
motion,  because  in  the  opposite  side  AC  there  is  always  a  part  HK, 
the  length  and  breadth  of  which  exactly  correspond  to  EF,  and  which 
suffers  an  equal  and  opposite  pressure.  But  if  EF  is  open,  and 
water  issues  from  it,  the  pressure  upon  HK  has  no  longer  a  counter- 
poise, and  hence  it  may  give  the  vessel  a  contrary  motion,  if  it  be 
easy  to  be  moved. 


The  Motions  of  Solid  Bodies  in  Liquids. 

275.  A  solid  body  cannot  move  in  a  liquid  without  putting  a  cer- 
tain quantity  of  its  mass  in  motion.  But  it  loses  just  as  much  of 
its  own  motion  as  it  communicates  to  the  liquid,  as  is  evident  from 
what  has  been  said. 

We  consider  this  loss  as  the  effect  of  a  force  which  the  liquid  op- 
poses to  the  body  put  in  motion,  and  call  it  the  resistance  of  the 
liquid.  The  efforts  of  the  greatest  mathematicians  have  not  yet  been 
able  to  reduce  the  theory  of  this  resistance  to  simple  and  exact  laws. 
Since  the  time  of  Newton  it  has  been  generally  admitted  that  this 


118  Liquid  Bodies. 

resistance  is  proportional  to  the  product  of  three  factors,  which  are 
the  square  of  the  velocity  of  the  body  in  motion,  the  extent  of  the 
surface  which  resists  this  velocity,  and  lastly,  the  density  of  the  • 
liquid,  supposing  all  other  circumstances  the  same  in  each  case.  But 
a  great  number  of  experiments  made  since  the  middle  of  the  last 
century,  principally  in  France,  have  proved  that  all  these  principles 
are  uncertain.  It  is  only  in  case  of  mean  velocities  that  they  agree 
tolerably  with  experiment.  When  the  velocities  are  very  great  or 
very  small,  they  deviate  very  widely.  What  has  been  said  of  the 
resistance  of  a  liquid  at  rest  may  also  be  applied  to  the  impulse  of  a 
liquid  in  motion  against  a  solid  body,  and  also  to  the  case  in  which 
both  have  motions  contrary  to  each  other. 

276.  We  come  now  to  consider  a  case  which  is  attended  with  no 
difficulty  ;  that  is,  the  vertical  descent  and  elevation  of  solid  bodies 
in  water. 

If  a  body  which  weighs  8  grains  displaces  only  7  grains  of  water, 
it  sinks.  Yet  as  its  mass  of  8  grains  is  put  in  motion  by  a  force  of 
only  one  grain,  it  would  fall  in  truth  with  a  velocity  uniformly  ac- 
celerated, if  the  water  made  no  resistance  ;  but  its  motion  would  be 
like  the  force  which  acts  upon  it,  8  times  less  than  in  a  vacuum ; 
moreover,  as  the  water  resists  it  in  its  descent,  its  acceleration  will 
be  weakened  at  each  moment ;  and  the  resistance  increasing  nearly 
as  the  square  of  the  velocity,  the  acceleration  will  diminish  very 
rapidly,  and  will  soon  become  nothing.  In  fact,  there  must  be  an 
instant  when  the  resistance  of  the  water  takes  from  the  body  just  as 
much  velocity  as  the  accelerating  force  of  gravity  communicates. 
After  this  moment  the  body  falls  with  a  perfectly  uniform  motion. 
This  moment  arrives  the  sooner  in  proportion  as  the  specific  gravity 
of  the  body  differs  less  from  that  of  water. 

Exactly  the  same  may  be  said  of  a  light  body  rising  in  water.  If 
the  liquid  made  no  resistance  it  would  ascend  with  a  uniformly  ac- 
celerated motion,  since  the  force  which  raises  it  is  constant.  But 
the  resistance  of  the  water  produces  precisely  the  same  effect  as  in 
the  last  case. 

In  a  transparent  vessel  these  two  kinds  of  motion  may  be  rendered 
visible  by  means  of  bodies  only  a  little  lighter  or  a  little  heavier  than 
water. 

277.  The  limits  of  an  elementary  work  do  not  permit  us  to  pre- 
sent any  thing  more  than  the  fundamental  principles  of  hydraulics. 
The  application  of  these  principles  to  the  great  variety  of  hydraulic 
engines  belongs  to  the  science  of  machines. 


SECTION  V. 

AERIFORM   BODIES. 


CHAPTER  XXVII. 

Elastic  Fluids  in  General. 

278.  IT  was  formerly  supposed  that  atmospheric  air  was  the  only 
elastic  fluid  in  nature.     Modern  chemistry  has  taught  us  that  there 
are  many  of  these  fluids  to  which  we  give  the  name  of  airs  or  gases. 
The  examination  of  gases  evidently  belongs  to  chemical  philosophy  ; 
and  accordingly  we  shall  only  present  those  views  of  the  subject 
which  are  indispensable  to  the  student  of  mechanical  philosophy. 

Atmospheric  Air. 

279.  It  is  principally  from  an  exact  observation  of  what  takes 
place  in  combustion,  that  we  learn  that  air  is  not  a  simple  substance, 
as  was  formerly  believed  ;  but  a  mixture  of  two  gases,  oxygen  and 
azote,  nearly  in  the  ratio  of  1  to  3.*     These  are  at  least  the  essen- 


*  More  exactly  a  volume  of  atmospheric  air  equal  to  1  contains 
0,21  of  oxygen ;  the  rest  is  a  mixture  not  yet  exactly  known,  of  azote 
and  carbonic  acid,  perhaps  also  of  some  other  gases.  The  most  pro- 
bable estimates  give  0,785  of  azote,  and  0,005  of  carbonic  acid ;  so 
that  azote  is  much  the  most  abundant.  It  does  not  contain  hydrogen 
in  sensible  quantities  ;  that  is,  we  cannot  admit  more  than  2  or  3  thou- 
sandths of  it.  These  proportions  of  atmospheric  air  are  exactly  the 
same  in  every  part  of  the  earth,  at  least  with  respect  to  the  oxygen 
it  contains.  Such  are  the  resu  ts  obtained  by  chemists,  and  princi- 
pally by  MM.  Humboldt  and  Gay-Lussac. 


120  Aeriform  Bodies. 

tial  ingredients  of  atmospheric  air;  but  we  should  be  deceived  if  we 
were  to  suppose  that  it  contains  nothing  but  these  substances.  Atmo- 
spheric air  has  the  very  active  property,  though  not  yet  sufficiently 
observed,  of  dissolving  most  of  the  fluids,  as  well  as  a  great  number 
of  solids,  and  of  communicating  its  elastic  state  to  portions  of  these 
bodies  more  or  less  considerable.  A  little  attention  to  the  phenom- 
ena which  occur  every  day  will  leave  no  doubt  on  this  point.  Thus 
every  body  which  diffuses  an  odour  must  be  in  fact  dissolved  by  air. 
Of  this  nature  are  most  metals,  lime,  moistened  clay,  &c.  But  air 
combines  also  with  many  inodorous  bodies  j  and  water  presents  a  very 
striking  proof  of  this.*  Moreover,  observation  proves  that  all  kinds 
of  gas,  especially  carbonic  acid  and  hydrogen,  are  naturally  pro- 
duced by  chemical  operations,  in  the  interior  of  the  earth,  or  at  its 
surface,  and  that  most  gases  combine  without  changing  their  state  of 
aggregation.  It  is  farther  evident,  that  millions  of  organized  beings 
live  and  die  in  atmospheric  air  ;  that  during  their  life  there  continu- 
ally takes  place  between  them  and  the  air  an  exchange  of  aliment  and 
secretions,  most  of  which  are  in  the  aeriform  state  ;  and  that,  during 
the  decomposition  of  these  beings,  their  constituent  principles  change 
into  simple  substances  more  or  less  elastic.  These  different  considera- 
tions are  sufficient  to  convince  us  that  atmospheric  air,  principally  in 
the  lower  regions,  is  a  combination  of  an  infinite  number  of  elastic 
fluids,  many  of  which  elude  not  only  our  senses,  but  also  the  most 
delicate  chemical  agents,  on  account  of  the  smallness  of  their  quan- 
tity. In  the  upper  regions  of  the  atmosphere,  the  air  appears  to  be 
more  simple  and  pure.  Yet  various  phenomena,  such  as  the  aurora 
borealis,  falling  stars,  meteors,  &c.,  which  the  simple  combination  of 


*  The  reasons  advanced  by  the  author  are  by  no  means  so  strong 
as  he  thinks.  It  appears  from  the  experiments  of  Saussure  and  Dai- 
ton,  that  the  evaporation  of  water  and  other  liquids  does  not  require 
for  its  production  the  action  of  a  dissolving  force,  for  it  takes  place 
with  equal  rapidity  in  a  vacuum.  It  is  probable  that  this  evapora- 
tion is  the  simple  effect  of  the  elastic  force  which  all  liquids  possess 
in  virtue  of  the  combined  caloric ;  and  air  by  its  material  presence 
and  pressure,  far  from  favouring  evaporation^  rather  presents  a  me- 
chanical obstacle,  and  forces  it  to  take  place  more  slowly.  Perhaps 
many  other  phenomena  of  the  same  kind,  in  which  bodies  are  re- 
duced to  vapour,  belong  also  to  internal  causes,  and  not  to  the  dis- 
solving force  of  air  or  the  gases.  But  it  is  not  necessary  to  pursue 
this  subject. 


Oxygen.  121 

the  two  essential  principles  of  air  could  not  produce,  prove  the  influ- 
ence of  other  substances  of  which  we  do  not  perhaps  suspect  the 
existence  in  these  elevated  regions.* 

We  shall  treat  in  separate  chapters  of  the  relations  of  air  to  water 
and  of  its  mechanical  properties.  ' 


Oxygen. 

280.  When  we  heat  strongly  the  oxyde  of  manganese  or  saltpetre, 
in  a  retort  exactly  closed,  there  is  disengaged,  especially  from  the 
first  of  these  substances,  a  considerable  quantity  of  air,  which  is 
almost  pure  oxygen.  We  find  in  works  on  chemistry  the  means  of 
obtaining  it  in  a  state  of  absolute  purity.  This  substance,  for  the 
discovery  of  which  we  are  indebted  to  Scheele  and  Priestley,  and  for 
the  exact  analysis  to  Lavoisier,  is  of  such  importance  in  nature,  that 
the  knowledge  of  it  is  almost  the  sole  cause  of  the  revolution  which 
has  taken  place  in  chemistry  within  the  last  30  years.  Without  oxy- 
gen, life  cannot  be  supported  ;  hence  it  is  sometimes  called  vital  air. 
It  is  necessary  to  combustion  ;  hence  Scheele  calls  it  air  of  Jire. 
It  enters  into  the  composition  of  most  of  the  substances  which  chem- 
ists call  acids,  and  on  this  account  Lavoisier  has  given  it  the  name 
of  oxygen,  that  is,  generator  of  acids.  The  term  dephlogisticated  air, 
employed  before  the  time  of  Lavoisier  was  derived  from  a  false 
theory,  and  ought  to  be  abandoned.  Oxygen  combines  not  only 
with  organic  inflammable  substances  and  with  most  saline  matter, 
but  also  with  many  inorganic  bodies,  and  particularly  with  the 
metals.  By  this  combination  it  takes  away  their  metallic  properties, 
and  changes  them  into  earthy  or  vitreous  substances  of  various  col- 
ours, called  metallic  oxydes,  metallic  earths,  metallic  calces.  The 
oxyde  of  manganese,  of  which  we  have  already  spoken,  and  the  sub- 


*  M.  Gay-Lussac  in  his  aerostatic  voyage  brought  air  from  the 
higher  regions  of  the  atmosphere;  and  this  air  presented  pre- 
cisely the  same  constituent  principles,  as  that  at  the  surface  of 
the  earth  ;  so  that  as  yet  there  is  nothing  to  prove  that  the  atmo- 
sphere is  not  throughout  of  the  same  nature  ;  for  the  phenomena 
which  we  do  not  yet  know  how  to  explain,  are  not  a  sufficient  reason 
for  admitting  the  existence  of  certain  substances  which  direct  experi- 
ment does  not  indicate. 

Elem.  16 


!22  Aertform  Bodies. 

stances  so  well  known  "under  the  names  of  rust,  verdegris,  white 
lead,  tin,  white  arsenic,  &c.,  belong  to  this  class.  Although  oxygen 
is  one  of  the  principal  parts  of  water,  since  it  makes  0,88  of  its 
mass,  water  absorbs  only  a  little  of  this  gas.* 

Instruments  have  been  invented,  called  eudiometers,  to  determine 
how  much  oxygen  atmospheric  air  contains  ;  and  it  appears  that  this 
quantity  is  constant.  The  construction,  as  well  as  use  of  these  in- 
struments belongs  entirely  to  chemistry. 


Azote. 

281.  When  we  burn  a  sufficient  quantity  of  phosphorus  in  the 
midst  of  a  certain  volume  of  atmospheric  air  completely  enclosed, 
about  a  quarter  of  this  volume  disappears,  and  what  remains  is  azote, 
a  gaseous  substance  not  respirable,  and  incapable  of  supporting  com- 
bustion. Although  azote  does  not  appear  to  enter  into  so  many  va- 
rious combinations  as  oxygen,  it  is  nevertheless  a  substance  of  ex- 
treme importance,  since  we  find  that  it  is  one  of  the  constituent 
principles  of  all  living  organic  bodies.  Some  German  philosophers 
call  it  saltpeterstoff  (substance  of  saltpetre)  because,  being  combined 
in  certain  proportions  with  oxygen,  it  produces  nitric  acid,  and  by 
combining  this  acid  with  potash,  we  obtain  saltpetre.  The  former 
denomination  of  dephlogisticated  air  is  to  be  rejected  entirely.  For 
further  particulars  respecting  this  substance,  the  reader  must  consult 
the  books  on  chemistry. f 


*  This  is  generally  true  ;  but  by  presenting  oxygen  to  water  at  the 
moment  when  it  is  disengaged  from  certain  combinations,  Thenard 
succeeded  in  making  it  absorb  more  than  200  times  its  volume  of 
this  gas,  with  a  degree  of  combination  so  intimate  that  even  the 
removal  of  the  atmospheric  pressure  did  not  effect  its  disengagement. 

t  It  is  a  singular  fact  that  almost  the  only  characteristics  by  which 
azote  can  be  known,  are  negative  ;  that  is,  we  only  know  that  it  does 
not  produce  such  and  such  effects.  The  only  exception  to  this  is  the 
property  discovered  by  Cavendish,  which  consists  in  the  power 
which  azote  possesses  of  forming  nitric  acid,  when  we  combine  it 
with  oxygen  by  means  of  the  electric  spark.  But  this  operation  is 
too  difficult  to  be  employed  as  a  common  test ;  so  that  if  there  exist 
in  azote,  as  is  very  possible,  several  distinct  substances  which  agree 
in  their  negative  properties,  they  may  easily  be  confounded. 


Hydrogen.  123 


Hydrogen. 

£8%.  Since  the  invention  of  balloons,  the  term  inflammable  air 
has  been  applied  generally  to  this  kind  of  gas,  which  in  a  pure  state 
is  12  or  13  times  lighter  than  atmospheric  air  of  the  same  elasticity. 
The  older  chemists  called  it  inflammable  spirit*  but  they  did  not 
carefully  examine  its  nature.  It  is  irrespirable.  No  combustion 
can  take  place  in  it,  although  it  becomes  itself  combustible  when 
combined  with  oxjgen.  When  we  mix  two  parts  of  this  gas  (mea- 
sured by  bulk  and  not  by  weight)  with  one  part  of  oxygen  or  four  of 
atmospheric  air,  we  obtain  what  is  called  detonating  gas.  We  have 
before  seen  that  the  inflammation  of  detonating  gas  produces  water. 
On  account  of  this  property  Lavoisier  gave  it  the  name  of  hydrogen. 
A  mass  of  water  is  composed  of  0,88  by  weight  of  oxygen,  and 
0,12  of  hydrogen.  This  gas  has  a  very  slight  affinity  for  water. 
We  obtain  it  in  its  pure  state  by  causing  the  vapour  of  water  to  pass 
through  an  iron  tube  heated  to  redness.  The  oxygen  of  the  water 
combines  with  the  iron,  and  the  hydrogen  passes  off.  We  obtain 
it  still  more  easily  by  dissolving  iron  or  zinc  in  diluted  muriatic  or 
sulphuric  acid.  Then  the  water  is  decomposed.  The  oxygen 
combines  with  the  metal  and  the  hydrogen  is  liberated. 


Carbonic  Add  Gas. 

283.  This  gas  was  formerly  called  Jived  air,  because  it  was  ori- 
ginally recognised  as  a  constituent  principle  of  many  solid  bodies, 
especially  of  calcareous  compounds.  It  forms  nearly  half  the  weight 
of  calcareous  spar,  marble,  limestone,  &,c.  This  gas  is  disengaged 
from  these  substances  by  pouring  upon  them  a  quantity  of  diluted 
sulphuric  or  other  acid.  It  has  since  been  discovered  that  this  air 
is  the  same  as  that  which  is  produced  by  the  combustion  of  char- 
coal, and  which  has  all  the  properties  of  an  acid  ;  hence  it  has 
received  the  name  of  carbonic  acid.  It  issues  in  great  quantities 
from  the  interior  of  the  earth  in  many  countries,  and  especially  in 
the  neighbourhood  of  volcanoes.  As  it  is  heavier  than  atmospheric 

*  In  German  this  is  called  brennbarer  Geist.  The  author  observes, 
that  perhaps  this  may  be  the  derivation  of  the  word  gas. 


124  Aeriform  Bodies. 

air,  and  only  mixes  very  slowly  with  it,  it  forms  in  some  places  a 
stratum  of  air  several  feet  thick,  in  which  no  animal  can  live,  because 
it  is  absolutely  irrespirable.  The  Grotto  del  Cane,  near  Naples, 
presents  a  phenomenon  of  this  kind.  If  we  mix  this  gas  with  water 
and  agitate  it  strongly,  it  will  take  into  combination  a  volume  nearly 
equal  to  its  own.  This  gas  will  even  hold  a  considerable  quantity  of 
water  in  solution.*  It  communicates  to  water  an  agreeable,  lively,  and 
acid  taste  ;  and  by  combining  with  it  in  different  proportions  it  forms 
the  essential  principle  of  mineral  waters.  Lime  water,  which  is 
made  by  dissolving  quick  lime  or  calcareous  earth  in  water,  furnishes 
a  convenient  method  of  discovering  its  presence  in  water.  When 
we  pour  a  small  quantity  of  such  a  liquid  into  lime  water,  the  latter 
becomes  turbid,  because  the  carbonic  acid  combines  with  the  lime, 
and  this  combination  is  insoluble  in  water. 

284.  Chemists  are  acquainted  with  many  other  gases,  and  new  ones 
are  discovered  from  time  to  time  ;  but  being  most  frequently  employ- 
ed for  their  chemical  properties  alone,  they  are  of  less  importance  to 
the  student  of  mechanical  philosophy  than  those  above  considered. 
All  these  substances  are  permanent  gases  ;  that  is,  they  Tetain  their 
aeriform  state  at  all  known  temperatures.  Gravity  and  elasticity  are 
mechanical  properties  common  to  them  all,  and  they  differ  in  differ- 
ent gases  only  in  intensity. 


Elastic  Vapours. 

285.  We  have  already  seen  in  the  section  on  Heat,  that  liquids 
may  be  made  to  pass  into  the  elastic  state  either  by  the  action  of 
heat,  or  by  the  dissolving  force  of  other  gases.  While  they  are  in 
this  state,  their  mechanical  properties  do  not  differ  essentially  from 
those  of  the  permanent  gases  ;  and  they  are  subject  to  the  same 
laws  of  equilibrium  and  motion.  Perhaps  even  the  difference  which 
exists  between  vapours  and  gases  is  as  little  essential  as  that  which  is 
found  to  take  place  between  liquid  mercury  and  the  solid  metals. 

*  Since  this  was  written  the  experiments  of  Dalton,  confirmed  by 
other  chemists,  prove  that  in  a  given  volume  of  carbonic  acid  there 
does  not  arise  a  greater  quantity  of  water  in  a  state  of  vapour  than 
in  any  other  gas. 


First  Principles  of  Hygrotnetry.  125 

CHAPTER  XXVIU. 

Water  in  Atmospheric  Air,  or  First  Principles  of  Hygrometry. 

286.  THE  mechanical  philosopher  must  necessarily  be  acquainted 
with  the  reciprocal  effects  of  air  and   water,  since  otherwise  we 
should  be  led  to  false  conclusions  in  many  circumstances  ;  for  exam- 
ple in  the  experiment  of  the  dilatation  of  gases  by  heat. 

Even  the  driest  air  always  contains  a  quantity  of  water ;  and  many 
instruments,  under  the  name  of  hygrometers,  have  been  invented  to 
measure  this  quantity ;  but  it  is  impossible  to  judge  exactly  of  the 
construction  and  use  of  these  instruments,  unless  we  know  the  laws 
according  to  which  water  distributes  itself  In  a  system  of  bodies,  all 
having  an  affinity  for  it.  We  must,  therefore,  explain  these  laws, 
though  they  belong  rather  to  chemistry  than  mechanics. 

287.  Water  may  be  contained  in  the  air  in  two  ways.     It  may 
float  in  it,  only  divided  into  very  small  bubbles,  without  actually  taking 
the  elastic  state  ;  or  it  may  be  perfectly  dissolved  in  it,  and  actually 
have  the  aeriform  state. 

288.  The  visible  vapour  which  rises  from  heated  liquids  is  formed 
of  small  bubbles  which  may  be  discovered  by  the  microscope.  These 
drops  would  fall  to  the  earth  in  perfectly  tranquil  air.    But.  it  is  diffi- 
cult to  find  a  mass  of  air  perfectly  at  rest,  and  the  slightest  motion  is 
sufficient  to  raise  a  great  quantity  of  these  drops.     If  only  a  few  are 
found  in  the  air  they  do  not  affect  its  transparency ;  but  still  they 
may  occasion  some  errors  in  the  results  of  experiments,  because  at 
the  least  elevation  of  temperature  they  may  pass  into  the  elastic 
state.     If  they  exist  in  considerable  quantities  they  form  visible  va- 
pour.    Hence   the   origin  of  fog  and  clouds.     Still  we  must  not 
conclude,  reciprocally,  that  all  visible  vapours  consist  of  bubbles  of 
water.    Not  only  may  all  other  liquids  form  visible  vapours,  but  solid 
bodies  may  also  do  it,  when  they  are  divided  into  portions  sufficiently 
attenuated.     The  vapour  or  smoke  of  flame  is  formed  solely  of  char- 
coal, minutely  divided  ;  and  the  white  vapour  produced  by  burning 
phosphorus  is  phosphoric  acid,  originally  solid,  but  now  in  a  state  of 
minute  division. 

289.  When  we  put  water  into  an  open  vessel  and  expose  it  to 
free  air,  it  gradually  diminishes  and  at  length  disappears,  because  it 
is  dissolved  in  air.     If  this  evaporation  takes  place  in  a  mass  of  air 


126  Aeriform  Bodies. 

which  is  accurately  enclosed  and  deprived  of  water,  its  volume  in- 
creases, and  its  elasticity  and  specific  gravity  are  changed.  This  is 
a  proof  that  the  evaporated  water  is  not  only  mixed  mechanically 
with  air,  but  that  it  is  also  chemically  combined  with  it,  and  conse- 
quently that  it  has  passed  to  the  elastic  state.  Not  only  atmospheric 
air,  but  perhaps  also  all  other  gases  without  exception,  may  combine 
in  this  manner  with  a  greater  or  less  quantity  of  water.  Air  does 
not  lose  its  transparency  on  account  of  the  water  dissolved  in  it ;  but 
while  in  this  state  it  may  even  appear  to  our  senses  very  dry.  This 
effect  is  reciprocal  between  air  and  water ;  and  the  parts  of  water 
which  are  not  yet  vapourized,  always  combine  with  some  particles  of 
air  to  which  they  communicate  their  state  of  aggregation,  that  is, 
cause  them  to  pass  to  the  liquid  state. 

^90.  The  dissolving  force  of  air  is  not  equally  great  under  all  cir- 
cumstances ;  heat  and  condensation  augment  it ;  cold  and  dilatation 
diminish  it.*  Thus  when  a  mass  of  air  has  absorbed  as  much  water 
as  it  can  contain,  if  it  is  cooled  or  dilated,  a  part  of  the  water  rendered 
elastic  takes  again  the  liquid  state  and  appears  in  bubbles  of  vapour. 
It  is  on  this  account  that  the  receiver  of  an  air-pump  is  often  covered 
with  vapour  when  the  air  becomes  rarefied  ;  and  it  is  for  the  same 
reason,  that  cold  bodies  become  moist  on  their  surface  when  brought 
into  hot  air. 

In  these  circumstances  the  water  is  said  to  be  precipitated.  On 
the  contrary,  the  bubbles  of  vapour  dissolve  or  change  into  elastic 
vapour  when  the  air  in  which  they  float,  becomes  heated  or  com- 


291.  Many  bodies,  independently  of  air,  have  a  great  affinity  for 
water.  When  a  body  of  this  kind  is  placed  in  a  mass  of  air  con- 
taining water  in  solution,  it  takes  from  this  air  a  part  of  its  water. 
The  more  water  it  has  already  attracted,  the  less  strongly  it  con- 
tinues to  attract ;  and  on  the  contrary,  the  more  water  the  air  has 
lost,  the  greater  is  the  force  with  which  it  retains  the  rest.  There 

*  Air  in  condensing  disengages  heat,  in  dilating  absorbs  it.  Ac- 
cordingly the  effects  of  condensation  and  dilatation  of  air  on  vapours 
are  to  be  referred  to  changes  of  temperature.  When  the  primitive 
temperature  is  restored,  the  quantity  of  vapour  capable  of  existing  in 
a  given  space,  becomes  exactly  the  same,  whatever  be  the  dilatation 
or  condensation  of  the  air  contained  in  this  space.  [See  additional 
note  at  the  end  of  the  chapter.] 


First  Principles  of  Hygrometry.  127 

must,  therefore,  necessarily  be  a  moment  when  both  bodies  retain 
the  water  with  equal  force ;  then  the  effect  ceases.  This  state  of 
rest  is  called  hygrometric  equilibrium.  If  a  mass  of  air  containing 
water  is  in  contact  with  different  bodies  of  this  kind,  each  of  them 
takes  from  it  a  part  of  its  water,  some  more  and  others  less,  accord- 
ing to  their  affinity  for  water ;  on  the  contrary,  if  bodies  which  have 
absorbed  water,  are  exposed  to  air  which  contains  less  water  than  is 
required  to  establish  the  hygrometric  equilibrium,  it  will  take  water 
from  them  till  this  equilibrium  is  effected. 

292.  There  probably  exists  no  body  which  has  not  some  affinity 
for   water ;  but  in  many  this  affinity  is  insensible.     Those  which 
show  the  greatest  affinity  for  this  liquid  are  called  hygrometric  bodies. 
To  this  class  belong  all  bodies  which  are  derived  from  organic  na- 
ture, as  wood,  bone,  ivory,  hair,  paper,  parchment,  the  epidermis 
which  covers  the  internal  and  external  parts  of  the  bodies  of  animals, 
musical  strings  made  of  it,  the  tubular  part  of  feathers,  silk,  &c. 
There  are  also  many  inorganic  bodies  which  are  hygrometric.     For 
example,  all  soluble  salts  remain  hygrometric  even  in  the  liquid  state, 
and  when  the  solution  is  saturated.     Most  of  the  acids,  and  espe- 
cially sulphuric  acid,  possess  this   property ;  also  slate,  argil,  and 
other  minerals  which  adhere  to  the  tongue.     We  may  also  reckon 
in  this  class  bodies  which  are  too  compact  to  imbibe  water,  but  the 
surface  of  which  becomes  covered  with  it  when  exposed  to  a  warm 
and  moist  air,  such  are  glasses,  metals,  &c. 

293.  As  the  temperature  and  density  of  the  atmosphere  are  con- 
tinually changing,  there  must  also  be  a  continual  exchange  of  water 
between  the  air  and  the  bodies  with  which  it  is  in  contact. 

294.  Such  are  the  observations  and   principles  upon  which  is 
founded  hygrometry,  or  the  estimation  of  the  quantity  of  water  con- 
tained in  the  atmosphere.     From  these  principles  it  will  be  easily 
inferred  that  water  in  distributing  itself  through  a  system  of  bodies, 
in  order  to  establish  the  hygrometric  equilibrium,  observes  laws  anal- 
ogous to  those  by  which  heat  is  propagated  in  order  to  produce  the 
thennometric  equilibrium.     Moreover  we  are  taught  by  chemistry, 
that  the  various  chemical  affinities  act  according  to  the  same  laws, 
which  are  general  for  all  substances.     This  is  a  decisive  reason  for 
admitting  the  .materiality  of  heat. 


128  Aeriform  Bodies. 


Addition  Relative  to  Hygrometry. 

295.  All  that  the  author  has  said  in  this  chapter  respecting  the  man- 
ner in  which  the  hygrometric  equilibrium  is  established  between  differ- 
ent substances  having  an  affinity  for  water,  is  perfectly  just ;  but  the 
evaporation  of  water  in  air  and  most  of  the  gases,  does  not  appear  to 
depend  upon  this  cause ;  for  experiments  prove  that  it  takes  place 
independently  of  affinity  ;  or  at  least  as  if  the  effect  of  affinity  were 
entirely  insensible. 

To  prove  the  truth  of  this  assertion,  we  must  call  to  mind  an  im- 
portant fact  which  Saussure,  and  after  him,  Volta  and  Dalton,  have 
established  by  very  exact  experiments,  which  is,  that  the  maximum 
of  elastic  vapour  which  can  arise  in  a  given  space,  depends  solely 
upon  the  temperature,  and  continues  invariable  when  the  tempera- 
ture remains  the  same  ;  whether  the  space  be  filled  with  air  of  any 
density,  or  be  a  vacuum.  Dalton  has  even  extended  this  fact  to  all 
the  gases  which  have  not  a  very  great  affinity  for  water ;  such  as 
oxygen,  azote,  and  hydrogen.  Some  restriction  is  perhaps  neces- 
sary for  carbonic  acid,  muriatic  acid,  and  ammoniacal  gas  ;  but  for 
the  rest,  and  especially  for  oxygen  and  azote,  which  are  the  elements 
of  atmospheric  air,  it  appears  very  evident  that  their  affinity  for  water 
does  not  produce  evaporation ;  for  then  this  affinity  would  be  the 
same  for  all,  which  is  hardly  probable ;  and  a  vacuum  would  act 
upon  water  with  an  equal  force  of  affinity,  which  is  absurd.  Besides, 
this  property  is  not  peculiar  to  the  vapour  of  water  ;  it  is  common  to 
all  evaporable  liquids,  as  alcohol,  ether,  ammonia,  muriatic  acid,  &C. 
Each  of  these  liquids  sends  off  a  determinate  quantity  of  vapour  in  a 
given  space,  when  the  temperature  is  the  same,  whether  this  space 
be  a  vacuum  or  be  filled  with  air  or  any  gas  whatever,  with  the  ex- 
ception of  those  cases  in  which  there  is  a  very  great  affinity. 

296.  According  to  this  single  principle,  which  is  founded  upon  exact 
and  rigorous  experiments,  the  whole  theory  of  hygrometry  becomes 
exceedingly  simple,  so  far  as  evaporation  is  concerned.  If  a  liquid 
is  exposed  freely  in  a  void  space,  or  one  which  is  filled  with  air,  a 
certain  quantity  will  evaporate,  depending  upon  the  dimensions  of  this 
space  and  upon  the  temperature.  This  quantity  may  be  measured 
by  its  weight,  and  by  the  pressure  which  its  elastic  force  produces 
upon  the  mercury  of  the  barometer.  If  the  space  is  indefinite  the 
liquid  will  be  entirely  evaporated ;  this  is  what  takes  place  in  the 


Hygrometry.  129 

open  air.  If  it  is  limited  the  evaporation  will  be  limited  also.  It 
will  cease  at  a  certain  limit,  depending  upon  the  dimensions  of  the 
space  and  the  temperature  ;  but  this  limit  will  be  the  same,  whether 
the  space  be  void  or  full  of  air.  Only  in  the  first  case  the  evapora- 
tion will  be  instantaneous,  because  nothing  opposes  it ;  in  the  second 
it  will  be  progressive  and  will  require  a  certain  interval  of  time,  on 
account  of  the  mechanical  obstruction  which  the  air,  by  its  presence, 
offers  to  the  dissemination  of  the  particles  of  the  liquid  ;  and  in  these 
two  cases,  after  a  greater  or  less  interval,  the  barometer  introduced 
into  this  space  will  indicate  the  same  increase  of  pressure. 

This  is  what  takes  place  with  respect  to  a  liquid  which  is  not  sub- 
jected to  any  foreign  force,  and  which  yields  solely  to  the  repulsive 
action  of  the  caloric  interposed  between  its  particles,  which  is  the 
determining  cause  of  evaporation.  But  if  the  liquid  is  retained  by 
a  solid  body,  which  has  an  affinity  for  it,  it  will  be  continually  acted 
upon  by  two  contrary  forces,  which  according  to  circumstances  may 
be  equal  or  unequal.  If  the  space  in  which  the  body  is  placed  be 
deprived  of  vapour,  the  elastic  action  will  have  all  its  energy,  and  a 
part  of  the  liquid  will  separate  from  the  solid  body,  taking  the  aeri- 
form state.  But  even  by  this  effect  the  preponderance  of  the  elas- 
tic force  will  be  found  to  be  diminished  ;  for  the  tendency  to  evapo- 
ration will  become  less  ;  and  on  the  contrary,  the  action  of  the  solid 
body  upon  the  water  which  remains  will  increase  in  proportion  to 
what  it  has  already  lost.  Hence  results  a  state  of  hygrometric  equi- 
librium; but  this  state  will  be  disturbed  by  a  change  of  temperature. 
If  the  temperature  be  raised  the  elastic  force  will  preponderate, 
and  a  new  quantity  of  liquid  will  evaporate.  If  it  be  lowered,  the 
affinity  of  the  body  will  preponderate,  and  a  portion  of  the  vapour 
being  absorbed  will  return  to  the  liquid  state.  These  constant 
changes  are  sufficiently  sensible  with  respect  to  certain  bodies,  as 
hair,  feathers,  cords,  to  vary  their  dimensions ;  and  we  may  thus 
observe  all  their  successions.  It  is  upon  this  property  that  the  in- 
struments called  hygrometers  are  founded ;  and  it  is  obvious  that  the 
action  of  these  instruments  admits  of  an  easy  explanation  upon  the 
principles  above  stated,  without  supposing  in  the  air  a  dissolving 
force  which  is  not  indicated  by  any  experiments.  The  whole  de- 
pends upon  the  simple  fact  of  the  moveable  equilibrium  between  the 
affinity  of  the  solid  body  for  water  and  the  elastic  force  of  heat. 

It  is  proper  to  observe  that  the  results  here  stated  were  not  gen- 
erally known  when  the  work  of  M.  Fischer  was  published  ;  other- 

Elem.  17 


130 


Aeriform  Bodies. 


wise  this  judicious  author  would  unquestionably  have  adopted  them. 
He  was  only  able  to  state  in  a  note  some  results  of  Dalton  which 
came  to  his  knowledge  while  the  work  was  in  the  press. 

To  supply  this  omission,  so  far  as  is  practicable  in  a  work  where 
correct  ideas  of  physical  phenomena  are  rather  to  be  stated  than 
carried  out  into  detail,  I  have  inserted  a  table  of  the  elastic  force  of 
aqueous  vapour  at  different  temperatures  from  20°  below  zero  of  the 
centesimal  scale  to  130°  above.  This  table  is  deduced,  by  interpo- 
lation, from  a  multitude  of  experiments  made  by  MM.  Dalton  and 
Gay-Lussac. 


Hygrometry. 


131 


Elastic  Force  of  Aqueous  Vapour  estimated  in  Millimetres  for  each 
Degree  of  the  Centesimal  Thermometer. 


Jegrees. 

'ension. 

Degrees. 

Tension. 

Degrees. 

Tension. 

Degrees. 

Tension. 

—  20 

,333 

18 

15,353 

56 

119,39 

94 

611,18 

—  19 

,429 

19 

16,288 

57 

125,31 

95 

634,27 

—  18 

,531 

20 

17,314 

58 

131,50 

96 

658,05 

—  17 

,638 

21 

18,317 

59 

137,94 

97 

682,59 

—  16 

,755 

22 

19,417 

60 

144,66 

98 

707,63 

—  15 

,879 

23 

20,577 

61 

151,70 

99 

733,46 

—  14 

2,011 

24 

21,805 

62 

158,96 

100 

760,00 

—  13 

2,152 

25 

23,090 

63 

166,56 

101 

787,27 

—  12 

2,302 

26 

24,452 

64 

174,47 

102 

815,26 

—  11 

2,461 

27 

25,881 

65 

182,71 

103 

843,98 

—  10 

2,631 

28 

27,390 

66 

191,27 

104 

873,44 

—  9 

2,812 

29 

29,045 

67 

200,18 

105 

903.64 

—  8 

3,005 

30 

30,643 

68 

209,44 

106 

934^81 

—  7 

3,210 

31 

32,410 

69 

219,06 

107 

966,31 

—  6 

3,428 

32 

34,261 

70 

229,07 

108 

994,79 

—  5 

3,660 

33 

36,188 

71 

239,45 

109 

1032,04 

—  4 

3,907 

34 

38,254 

72 

250,23 

110 

1066,06 

—  3 

4,170 

35 

40,404 

73 

261,43 

111 

1100,87 

—  2 

4,448 

36 

42,743 

74 

273,03 

112 

1136,43 

—  1 

4,745 

37 

45,038 

75 

285,07 

113 

1172,78 

0 

5,059 

38 

47,579 

76 

297,57 

114 

1209,90 

1 

5,393 

39 

50,147 

77 

310,49 

115 

1247,81 

2 

5,748 

40 

52,998 

78 

323,89 

116 

1286,51 

3 

6,123 

41 

55,772 

79 

337,76 

117 

1325,98 

4 

6,523 

42 

58,792 

80 

352,08 

118 

1366,22 

5 

6,947 

43 

61,958 

81 

367,00 

119 

1407,24 

6 

7,396 

44 

65,627 

82. 

382,38 

120 

1448,83 

7 

7,871 

45 

68,751 

83 

393,28 

121 

1491,58 

8 

8,375 

46 

72,393 

84 

414,73 

122 

1534,89 

9 

8,909 

47 

76,205 

85 

431,71 

123 

1  578,96 

10 

9,475 

48 

80,195 

86 

449,26 

124 

1623,67 

11 

10,074 

49 

84,370 

87 

467,38 

125 

1669,31 

12 

10,707 

50 

88,742 

88 

486,09 

126 

1715,58 

13 

1  ,378 

51 

93,301 

89 

505,38 

127 

1762,56 

14 

12,087 

52 

98,075 

90 

525,28 

128 

1810,25 

15 

12,837 

53 

103,06 

91 

545,80 

129 

1858,63 

16 

13,630 

54 

108,27 

92 

566,95 

130 

1907,67 

17 

14,468 

55 

113,71 

93 

588,74 

132  Aeriform  Bodies. 

297.  Trusting  to  our  senses,  we  judge  as  erroneously  of  humidity 
as  of  heat.    We  consider  air  or  any  other  substance  as  moist,  when  it 
deposites  moisture  upon  our  bodies.  We  judge  that  it  is  dry  when  it 
takes  away  moisture.     The  same  mass  of  air  may,  therefore,  appear 
dry  to  one  observer  and  moist  to  another.     For  this  reason  philoso- 
phers directed  their  attention  long   ago  to  the  construction  of  an 
instrument  which  should  indicate  the  humidity  of  the  atmosphere 
with  greater  certainty  than  the  sense  of  touch.     We  have  not  room 
to  describe  the  numerous  attempts  of  this  kind  that  have  been  made. 
The  first  instruments  were  extremely  defective,  and  even  the  best  at 
the  present  day  are  far  from  approaching  the  accuracy  of  the  ther- 
mometer.  It  is  a  singular  fact  that  we  are  much  better  able  to  mea- 
sure a  substance  that  is  imperceptible  to  our  senses,  than  one  which 
we  can  directly  observe. 

298.  We  shall  only  remark  with  respect  to  die  earliest  hygrome- 
ters, that  the  best  of  them  are  founded  upon  the  hygrometric  proper- 
ties of  cat-gut  cords,  which  untwist  by  the  effect  of  the  humidity 
which  they  acquire,  and  thus  become  shorter  since  they  augment  in 
size. 

299.  Among  the  instruments  of  this  kind  lately  invented,  there 
are  only  two  which  deserve  the  name  of  hygrometers  ;  that  of  Saus- 
sure,  and  still  more  recently  that  of  Deluc.     We  shall  only  speak 
of  the  first,  which  is  most  in  use.     The  hygrometric  body  employed 
by  Saussure,  is  a  hair,  deprived  of  all  oily  substance  by  being  boiled 
in  a  weak  solution  of  potash.   The  hair,  thus  prepared,  contracts  by 
dryness,  and  is  lengthened  by  humidity.     It  is  firmly  attached  at 
one  of  its  extremities ;  the  other  end  is  fixed  to  an  index  very  easily 
moved,  which  is  turned  one  way  by  the  hair,  and  the  other  by  a 
small  weight.     This  index  by  its  movement  on  a  graduated  arc, 
indicates  the  contractions   or    elongations   which  the   hair   under- 
goes in  consequence  of  the  variations  of  humidity  in  the  surrounding 
air. 

300.  The  essential  advantage  of  this  hygrometer  consists  in  this, 
that  we  determine  by  experiment  two  fixed  points,  those  of  extreme 
dryness  and  extreme  moisture,  that  serve  as  the  limits  of  scales 
which  admit  of  being  compared  with  one  another.     Lambert  had 
already  conceived  the  idea  of  this  instrument,  but  he  did  not  succeed 
in  the  execution  of  it  so  well  as  Saussure  and  Deluc. 

301.  We  determine  the  point  of  extreme  dryness  by  placing  the 
instrument  under  a  large  glass  receiver  together  with  calcined  salts, 


General  Remarks  upon  Hygrometry.  133 

.and  letting  it  remain  in  this  situation  as  long  as  we  can  perceive  that 
the  hair  continues  to  contract. 

202.  We  determine  the  point  of  extreme  humidity  by  suspending 
the  instrument  under  a  receiver,  the  sides  of  which  are  wet  with 
water.  The  receiver  itself  is  placed  on  a  shelf  covered  with  water 
to  prevent  the  introduction  of  external  air.  The  apparatus  thus  dis- 
posed is  suffered  to  remain  as  long  as  the  hair  continues  to  lengthen, 
and  we  note  the  point  on  the  graduated  arc  at  which  this  limit  takes 
place. 

The  distance  between  these  two  fixed  points  are  divided  into  100 
parts  called  degrees. 

General  Remarks  upon  Hygrometry. 

303.  It  may  be  asked  what  a  hygrometer  properly  indicates. 
According  to  the  theory  explained  above,  the  elongation  of  the  hair 
indicates  that  it  has  received  water  from  the  air  ;  and  the  shortening 
that  it  has  parted  with  water  to  the  air ;  and  a  state  of  rest,  that  the 
hair  and  the  atmosphere  are  in  a  state  of  hygrometric  equilibrium. 
Consequently,  if  the   forces  with   which  the  air  and  hygrometric 
bodies  attract  water  were  in  constant  ratios  to  each  other,  the  mo- 
tion of  the  index  would  be  affected  only  by  the  augmentation  or 
diminution  of  water  contained  in  the  atmosphere,  and  there  would 
be  no  difficulty  in  determining  in  what  ratios  it  existed  for  each  de- 
gree of  the  hygrometer.     But  as  the  elastic  force  of  the  aqueous 
vapour  increases  when  the  temperature   is   raised,  and    decreases 
when  it  is  lowered,  the  index  of  the  hygrometer  must  move,  although 
the  absolute  quantity  of  vapour  does  not  change  whenever  a  change 
of  temperature  occurs.     Moreover,  as  the  air  may  contain,  inde- 
pendently of  water,  a  mixture  of  many  other  substances,  all  of  which 
act  upon  the  water  with  a  particular  force,  it  is  evident  that  the  indi- 
cation given  by  a  hygrometer,  is  a  complicated  result  of  many  forces. 
Jt  happens  also  that  liquid  water  suspended  in  the  air,  and  the  bubbles 
of  vapour  which  float  in  it,  act  conjointly  upon  the  hygrometer,  with- 
out our  being  able  to  distinguish  them.     These  facts  hardly  permit 
us  to  cherish  the  hope  of  giving  to  these  instruments  that  degree  of 
accuracy  which  is  desirable. 

304.  Another  defect  of  almost  all  the  hygrometers  hitherto  invent- 
ed, consists  in  this,  that  the  hygrometric  substance  is  of  organic 
origin.     It  is  trim;  bodies  of  this  kind  are  for  the  most  part  very 


134  Aeriform  Bodies. 

sensible  to  moisture  ;  but  it  is  a  general  law  that  every  body  pro- 
duced by  an  organic  force,  must,  after  this  force  is  destroyed,  change 
its  chemical  constitution  by  being  exposed  to  the  air,  to  moisture, 
and  to  multiplied  variations  of  temperature.  These  hygrometric 
substances  must,  therefore,  in  time  become  unfit  to  be  employed, 
since  in  changing  their  material  properties  they  change  also  their 
attractive  force  with  respect  to  water.  This  is  a  circumstance  to 
which  no  attention  seems  hitherto  to  have  been  paid. 

305.  As  there  is  no  probability  that  we  can  ever  obtain  a  hygrom- 
eter the  scale  of  which  shall  indicate  immediately  how  many  parts 
of  water  are  contained  in  the  air,  there  remains  no  other  means  of 
making  this  estimate  exactly,  than  by  chemical  decompositions.  Cal- 
cined salts  furnish  a  method  sufficiently  convenient  and  precise  for 
this  purpose.     It  is  necessary  to  put  the  air  to  be  examined  in  a  ves- 
sel of  a  capacity  accurately  known,  and  expose  it  thus  for  a  long 
time  to  the  action  of  calcined  salts ;  taking  great  care  to  prevent 
every  approach  of  moisture.     The  increase  of  weight  in  the  salt, 
determined  by  a  very  delicate  balance,  will  express  the  quantity  of 
water  contained  in  the  air ;  only  this  estimate  will  be  a  little  too 
small,  since  it  is  evident  from  the  theory  we  have  described,  that  no 
body  can  take  from  the  air  all  the  water  which  it  contains. 

Addition. 

306.  Since  the  publication  of  this  work,  M.  Gay-Lussac  has  found 
out  a  very  simple  process  for  measuring  the  actual  quantities  of  aque- 
ous vapour  corresponding  to  the  indications  of  Saussure's  hygrometer. 
This  process  consists  in  enclosing  the  hygrometer  in  a  large  glass 
vessel  partly  filled  with  water,  or  a  known  saline  solution,  and  the 
tension  of  which  in  the  barometer  at  a  known  temperature  has  been 
previously  estimated.     After  having  carefully  closed  all  communica- 
tion between  the  interior  of  the  vessel  and  the  external  air,  the  appa- 
ratus is  left  to  itself  for  several  days.     The  liquid  at  length  saturates 
as  far  as  its  force  of  emission  admits,  the  interior  of  the  vessel 
with  aqueous  vapour,  and  the  hygrometer  being  placed  in  equili- 
brium with  it,  at  length  stops  at  a  certain  degree  of  its  division. 
We  thus  learn  that  this  degree  corresponds  to  the  known  tension 
of  the  liquid,  and  consequently  to  the  quantity  of  vapour  with  which 
we  know  this  tension  must  fill  the  space.     By  repeating  the  ex- 
periment with  various  liquids  of  different  tension*,  from  pure  water 


Hygrometry. 


135 


which  produces  complete  saturation,  to  a  drying  liquid  such  as  sul- 
phuric acid,  which  produces  extreme  dryness,  we  obtain  a  succes- 
sion of  results  which  being  interpolated,  express  the  general  law  ap- 
plicable to  the  intermediate  degrees.  In  this  manner  the  following 
table  was  formed  from  the  results  obtained  by  M.  Gay-Lussac,  and 
kindly  communicated  to  me. 

Hygrometric  Table,  constructed  for  the  temperature  of  10  centesi- 
mal degrees,  from  the  experiments  of  M.  Gay-Lussac. 


'ension 
of  the 

)egrees  of 
the  Hair 

Tension 

of  the 

Degrees  of 
the  Hair 

Tension 
of  the 

Degrees  of 
the  Hair 

Tension 
of  the 

Degrees  of 
the  Hair 

"apour. 

Hygrom- 

'apour. 

Hygrom- 

Vapour. 

Hygrom- 

Vapour. 

Hygrom- 

eter. 

eter. 

eter. 

eter. 

0 

0,00 

26 

47,55 

52 

73,68 

78 

89,51 

1 

2,19 

27 

48,86 

53 

74,41 

79 

90,03 

2 

4,37 

28 

50,18 

54 

75,14 

80 

90,55 

3 

6,56 

29 

51,49 

55 

75,87 

81 

91,05 

4 

8,75 

30 

52,81 

56 

76,54 

82 

91,55 

5 

10,94 

31 

53,96 

57 

77,21 

83 

92,05 

6 

12,93 

32 

55,11 

58 

77,88 

84 

92,54 

7 

14,92 

33 

56,27 

59 

78,55 

85 

93,04 

8 

16,92 

34 

57,42 

60 

79,22 

86 

93,52 

9 

18,91 

35 

58,58 

61 

79,84 

87 

94,00 

10 

20,91 

36 

59,61 

62 

80,46 

88 

94,48 

11 

22,81 

37 

60,64 

63 

81,08 

89 

94,95 

12 

24,71 

38  1  61,66 

64 

81,70 

90 

95,43 

13 

26,61 

39 

62,69 

65 

82,32 

91 

95,90 

14 

28,51 

40 

63,72 

66 

82,90 

92 

96,36 

15 

30,41 

41 

64,63 

67 

83,48 

93 

96,82 

16 

32,08 

42 

65,53 

68 

84,06 

94 

97,29 

17 

33,76 

43 

66,43 

69 

84,64 

95 

97,75 

18 

35,43 

44 

67,34 

70 

85,22 

96 

98,20 

19 

37,11 

45 

68,24 

71 

85,77 

97 

98,69 

20 

38,78 

46 

69,03 

72 

86,31 

98 

99,10 

21 

40,27 

47 

69,83 

73 

86,86 

99 

99,55 

22 

41,76 

48 

70,62 

74 

87,41 

100 

100,00 

23 

43,26 

49 

71,42 

75 

87,95 

24 

44,75 

50 

72,21 

76 

88,47 

25 

46,24 

51 

72,94 

77 

88,99 

This  table  is  constructed  for  the  purpose  of  giving  the  degrees  of  the  hair 
hygrometer  when  the  tension  of  aqueous  vapour,  actually  existing  in  the 

air,  is  known.    The  tension  of  aqueous  vapour  for  the  state  of  complete 

saturation,  is  represented  by  100,  and  the  other  smaller  tensions  are  ex- 

pressed in  centesimal  parts  of  this  unit.    Consequently,  if  we  observe  them 
under  another  form,  for  example,  in  millimetres,  it  is  necessary,  in  order  to 

apply  them  to  our  table,  to  multiply  by  100  afcd  to  divide  by  9,475  mil- 

limetres, which,  from  the  table  of  page  131,  expresses  the  total  tension 

of  vapour  in  millimetres  at  the  temperature  of  10°  centesimal. 

130 


.leriform  Bodies 


Hygrometric  Table,  constructed  for  the  Temperature  of  10  centesimal 
degrees,  from  the  experiments  of  M.  Gay-Lussac. 


Degrees  of 
the  Hair 
Tveroroe- 

Tension. 

Degrees 
of  the 
Hair  Hy- 

Tension. 

Degrees 
of  the 
Hair  Hy- 

'ension. 

Degrees 
of  the 
Hair  Hy- 

Tension. 

"Jo1"1" 

ter. 

grometer. 

rometer 

ronneter. 

o 

0,00 

26 

12,59 

52 

29,38 

78 

58,24 

1 

0,45 

27 

13,14 

53 

30,17 

79 

59,73 

2 

0,90 

28 

13,69 

54 

30,97 

80 

61,22 

3 

1,35 

29 

14,23 

55 

31,76 

81 

62,89 

4 

1,80 

30 

14,78 

56 

32,66 

82 

64,57 

5 

2,25 

31 

15,36 

57 

33,57 

83 

66,24 

6 

2,71 

32 

15,94 

58 

34,47 

84 

67,92 

7 

3,18 

33 

16,52 

59 

35.37 

85 

69,59 

8 

3,64 

34 

17,10 

60 

36^28 

86 

71,49 

9 

4,10 

35 

17,68 

61 

37,31 

87 

73,39 

10 

4,57 

36 

18,30 

62 

38,34 

88 

75,29 

11 

5,05 

37 

18,92 

63 

39,36 

89 

77,19 

12 

5,52 

38 

19,54 

64 

40,39 

90 

79,09 

13 

6,00 

39 

20,16 

65 

41,42 

91 

81,09 

14 

6,48 

40 

20,78 

66 

42,58 

(M 

83,08 

15 

6,96 

41 

21,45 

67 

43,73 

93 

85,08 

16 

7,46 

42 

22,12 

68 

44,89 

94 

87,07 

17 

7,95 

43 

22,79 

69 

46,04 

95 

89,06 

18 

8,45 

44 

23,46 

70 

47,19 

96 

91,25 

19 

8,95 

45 

24,13 

71 

48,51 

97 

93,44 

20 

9,45 

46 

24,86 

72 

49,82 

98 

95,63 

21 

9,97 

47 

25,59 

73 

51,14 

99 

97,81 

22 

10,49 

48 

26,32 

74 

52,45 

100 

100,00 

23 

11,01 

49 

27,06 

75 

53,76 

24 

11,53 

50 

27,79 

76 

55,25 

25 

12,05 

51 

28,58 

77 

56,74 

Tliis  table  is  constructed  for  the  purpose  ol  giving  the  tensions  of  \  .ipour 
answering  to  the  degrees  of  the  hygrometer.    These  tensions  are,  as  in  the 
preceding  table,  expressed  in  centesimal  parts  of  the  total  tension.     Conse- 
quently, if  we  would  express  them  in  millimetres,  when  the  degree  of  the 
hygrometer  should  have  indicated  them,  it  is  necessary  to  multiply  them 
by  9,475  millimetres,  and  to  take  the  hundredth  part  of  the  product. 

Barometer.  137 


CHAPTER  XXIX. 

Barometer  and  Mr-Pump. 

307.  WE  now  proceed  to  examine  more  particularly  the  mechan- 
ical properties  of  air.  For  this  purpose  we  must  first  be  made  ac- 
quainted with  two  instruments,  which  are  of  great  importance  in  natu- 
ral science,  and  the  proper  design  of  which  is  to  make  known  the 
mechanical  properties  of  the  air,  that  is,  its  gravity  and  dilatability. 
These  instruments  are  called  the  barometer  and  air-pump. 


Barometer. 

308.  We  fill  with  mercury  a  glass  tube  AE  (fg.  46),  the  length 
of  which  should  exceed  30  inches,  and  the  diameter  of  the  bore  be 
at  least  TV  of  an  inch,  one  of  its   extremities  A  being    hermeti- 
cally sealed.     We  then  close  with  the  finger  the  orifice  B  of  the 
tube,  invert  it,  and  immerse  this  extremity  in  a  vessel  of  mercury 
CD,  taking  care  that  no  air  enters.     Then,  if  we  remove  the  finger 
which  closes  the  orifice,  the  mercury  will  descend  in  the  tube  ;  but 
not  to  the  level  CD  of  the  vessel ;  it  will  remain  at  an  elevation  EF 
of  about  30  inches.     If  tbe  surface  of  the  mercury  CD  were  not 
exposed  to  any  pressure,  it  would  descend  to  E,  according  to  the 
laws  of  hydrostatics.     The  column  of  mercury,  therefore,  can  only 
be  sustained  by  the  pressure  of  the  atmosphere  upon  the  free  surface 
CD  of  the  mercury.     This  experiment,  which  Torricelli  first  made 
at  Florence,  in  1644,  not  only  serves  to  prove  that  the  air  exerts  a 
pressure,  but  also  indicates  the  exact  measure  of  this  pressure  ;  for 
we  see  that  it  is  just  equivalent  to  that  of  a  column  of  mercury  of 
the  height  EF.     When  the  operation  is  performed  with  proper  care 
there  is  in  the  tube  above  the  point  F  a  space  entirely  void  of  air. 
This  is  called  the  Torricellian  vacuum.     The  entire  apparatus  is 
called  the  Torricellian  tube,  and  when  it  is  furnished  with  a  scale  to 
measure  the  height  EF,  it  takes  the  name  of  barometer. 

309.  Various  experiments  have  been  made  for  the  purpose  of 
ascertaining  the  best  form  to  be  given  to  the  tube.     The  most  sim- 
ple and  advantageous  arrangements  are  represented  in  figures  47, 
48,  49.    Figure  47  represents  a  bason  barometer.  GBH  is  a  vessel 

Elem.  18 


13&  .ileriform  tfodies. 

of  wood  or  glass  attached  to  the  tube  BH.  There  may  be  at  Um 
place  a  small  opening  to  facilitate  the  passage  of  the  air  into  the  inte- 
rior of  the  vessel,  though  it  penetrates  very  easily  even  through 
compact  wood.  Figure  48, 'represents  the  phial  barometer,  so  call- 
ed from  the  form  of  the  tube.  Figure  41),  represents  a  barometer 
consisting  of  a  single  tube  JlBG  of  a  magnitude  as  uniform  as  possi- 
ble ;  it  is  called  the  syphon  barometer.  This  last  is  the  instrument 
most  used  for  experiments.*  It  is  very  obvious  that  the  scale  divid- 
ed into  inches,  which  is  annexed  to  each  of  these  barometers  to 
measure  the  height  of  the  column  EF,  must  be  made  with  very  great 
accuracy.  One  general  condition  of  every  good  barometer,  is,  that 
the  space  JLF  should  be  free  of  air,  and  moreover,  that  the  interior 
diameter  of  the  tube  should  be  at  least  one  tenth  of  an  inch ;  for 
where  it  is  less,  the  mercury  remains  too  low,  even  when  there  is  a 
complete  vacuum,  on  account  of  the  capillary  attraction. f 

310.  Soon  after  the  invention  of  the  barometer,  it  was  observed 
that  the  pressure  of  the  air  is  variable,  and  that  the  mercury  rises  or 
falls  about  one  inch  above  and  below  its  mean  height  at  F.  It  was 
also  observed  that  there  is  a  certain  relation  between  the  state  of  the 
barometer  and  the  state  of  the  weather ;  since,  in  fact,  when  the 
mercury  in  the  barometer  is  high,  the  weather  is  ordinarily  serene  ; 
and  it  becomes  variable  when  the  mercury  is  low.  But  this  rule  is 
not  certain,  though  it  is  verified  more  frequently  than  it  (ails.  A 
more  particular  explanation  of  this  must  be  sought  in  physical  geo- 
graphy. 


*  Its  principle  advantage  consists  in  being  independent  of  the 
effects  of  capillary  attraction.  If  the  tube  is  sensibly  of  a  uniform 
bore  in  its  two  branches,  the  convexity  of  the  surface  of  the  mercury 
produces  an  equal  action  in  both,  which  does  not  disturb  the  equili- 
brium, and  the  weight  of  the  atmosphere  is  exactly  represented  by 
the  difference  of  height  of  the  two  columns. 

t  This  is  true  for  ordinary  barometers  ;  but  the  interior  of  the 
tube  may  be  so  well  dried  by  repeated  boiling,  that  the  surface  of 
the  mercury  will  be  plane,  and  even  concave.  Then  the  liquid  will 
be  above  the  level  and  not  below  it.  This  remark  is  simply  theoreti- 
cal; for,  in  practice,  if  the  surface  of  the  mercury  were  concave  in- 
stead of  being  convex,  there  would  be  an  opposite  inconvenience. 
In  general  it  is  best  to  avoid  havhjg  narrow  tubes,  on  account  of  ca- 
pillary attraction. 


Barometer.  139 

The  ordinary  barometers  of  the  shops,  intended  only  to  indicate 
ihe  state  of  the  atmosphere,  without  a  scale  graduated  in  inches,  are 
unfit  for  a  scientific  observer.  They  serve  Vvell  the  purpose  to  which 
they  are  applied,  which  is  to  indicate  the  variations  of  atmospheric 
pressure.  Indeed,  if  we  take  for  the  point  of  departure  the  mean 
height  of  the  barometer  at  the  place  where  it  is  used  ;  when  it  is 
above  this  height  the  weather  is  commonly  serene  and  constant ;  and 
when  below,  it  is  almost  always  variable. 

But  the  variations  in  the  state  of  the  barometer  are  not  the  same 
throughout  the  earth.  At  the  equator  and  on  high  mountains  the 
variations  are  very  small.  Its  changes  become  more  considera- 
ble as  we  approach  the  poles,  and  particularly  in  low  countries. 
Illustrations  of  this  phenomenon  belong  to  physical  geography. 

311.  Shortly  after  the  invention  of  the  barometer  it  was  observed 
that  the  mercury  descends  when  the  instrument  is  carried  to  a  more 
elevated  situation.     Indeed  it  descends  about  one  tenth  of  an  inch 
for  87  feet.     From  this  observation  we  may  compare  the  gravity  of 
the  air  with  that  of  mercury  or  water ;  for  one  tenth  of  an  inch  of 
mercury  exerts  the  same  pressure  as  87  feet  of  air ;  and  as  87 
feet  =.  10440  tenths  of  an  inch,  it  will  be  seen  how  many  times 
mercury  is  heavier  than  air.    If  we  divide  this  number  by  13,57,  the 
specific  gravity  of  mercury,  the  quotient  769  indicates  how  many 
times  water  is  heavier  than  air.*     This  observation  has  also  given 
rise  to  the  ingenious  idea  of  measuring,  altitudes  by   the   barom- 
eter.     The  principle  upon  which  this  problem  depends  and  the 
manner  of  performing  it,  will  be  explained  hereafter. 

312.  We  may  determine  exactly  by  the  barometer  the  pressure 
which  the  air  exerts  upon  a  given  surface.     The  following  calcula- 
tion furnishes  an  estimate,  which  may,  at  the  same  time,  show  how  a 
more  exact  one  is  to  be  obtained.     When  the  barometer  is  at  30 
inches,  the  air  presses  upon  the  surface  of  a  square  inch  as  much  as 
a  column  of  mercury,  having  a  square  inch  for  its  base,  and  30  in- 
ches for  its  altitude.   This  column  comprehends,  therefore,  30  cubic 
inches.     Now  as  a  cubic  inch  of  water  is  equal  to  252,525  grains, 
30  cubic  inches  of  water  =  30  X  252,525  grains,  or   15,78  troy 
ounces.     Whence  mercury  being  13,57  times  heavier  than  water, 
30  cubic  inches  of  mercury   =  15,78    X  13,57  or  214,12  troy 

*  M.  Arago  and  myself  found,  by  careful  experiment,  that  at  the 
temperature  of  melting  ice  and  under  the  pressure  29,92  inches,  the 
weight  of  the  air  is  to  that  of  water  as  1  to  77O. 


140  Aeriform  Bodies. 

ounces.  This  is  equal  to  234,7  ounces  avoirdupois,  or  to  14,7  lb. 
We  infer,  therefore,  that  the  pressure  of  the  air  amounts  to  nearly 
15  lb.  upon  every  square  inch,  or  to  about  one  ton  upon  every  square 
foot. 

313.  If  we  would  construct  a  barometer  with  water  instead  of 
mercury,  it  would  be  necessary  to  have  it  nearly  14  times  as  long  ; 
that  is,  between  33  and  34  feet.     But  for  other  reasons,  a  water 
barometer  would  be  a  very  inconvenient  instrument.     Still  it  is  im- 
portant to  know  to  what  height  water  may  be  raised  by  the  pressure 
of  the  air  ;  and  hence  it  will  be  seen  why  the  exhausting  part  of  all 
kinds  of  pumps  must  not  be  more  than  33  feet  in  height. 

314.  For  very  exact  barometric  observations,  it  is  necessary  to 
apply  a  small  correction  on  account  of  heat ;  for  since  heat  dilates 
the  mercury,  it  is  manifest  that,  the  pressure  remaining  the  same,  if 
the  mercury  is  heated  the  column  will  be  lengthened,  and  its  highest 
point  will  be  elevated  in  the  tube  which  sustains  it.    We  should  have 
no  need  of  any  correction  if  it  were  possible  always  to  preserve  the 
mercury  at  the  same  temperature,  for  example  at  32°  ;  but  since 
this  is  impossible,  it  is  necessary  to  reduce  the  different  elevations  to 
the  same  temperature  by  calculation.     We  commonly  select  for  this 
purpose  the  temperature  of  32°.     According  to  the  best  observations 
mercury  expands  77*4  ?,  or  0,0001  nearly  for  1°  of  Fahrenheit's 
thermometer.     Thus  to  reduce  a  given  height  of  the  mercury  to  the 
temperature  of  32°,  we  must  subtract  for  each  degree  above  32° 
0,0001  of  the  whole  height  of  the  column,  and  add  as  much  for 
each  degree  below  32°.     To  have  the  temperature  of  the  mercury 
as  precisely  as  possible,  we  attach  a  thermometer  to  the  frame  work 
of  the  barometer.* 

*  Strictly  speaking,  when  the  temperature  of  the  mercury  is  t  de- 
grees above  32°,  we  ought,  in  order  to  reduce  it  to  32°,  to  subtract, 

not  9TJ  2>  but  9^42    i  ~t  of  its  length,  for  each  degree ;  and  when  the 

temperature  is  t  degrees  below  32°,  we  should  add for  each 

9742  —  t 

degree.  For  let  /  be  the  length  of  the  column  at  32°,  and  I'  its 
length  at  t  degrees  ;  we  have 

1  +  9742^  =  l>> 

whence  I  = __ =  //  __ 

1    , t__  9742 -f? 

r9742 


Air-Pump.  141 


Mr-Pump. 

315.  In  the  year  1650,  Guericke,  of  Magdeburg,  invented  one  of 
the  most  important  instruments  that  is  known  in  philosophy,  the  air- 
pump,  by  means  of  which  we  can  remove  the  air  from  the  interior  of 
a  vessel,  or  at  least  rarify  it  to  a  very  great  degree.  We  have  not 
room  for  a  description  of  its  original  construction,  and  the  changes 
which  it  has  undergone.  We  shall  only  indicate  here  the  essential 
parts  of  the  most  simple  apparatus  of  this  kind.  ABCD  (fg.  50) 
is  a  hollow  cylinder  of  metal,  the  interior  of  which  must  be  made 
with  great  exactness.  The  piston  EF  admits  of  being  raised  and 
depressed  in  the  interior  of  the  cylinder,  by  means  of  the  rod  G, 
without  suffering  the  air  to  enter.  The  piston  is  pierced  in  the  mid- 
dle HI,  by  an  aperture.  A  piece  of  gummed  taffeta  is  stretched 
over  the  orifice  at  H,  and  confined  at  its  two  extremities  ;  so  that  the 
air  which  comes  from  below  through  HI,  will  raise  it  and  escape  ;  but 
the  air  from  above  presses  it  against  the  opening  and  closes  the  passage. 
This  apparatus  attached  to  the  piston  is  called  a  valve.  In  the  bot- 
tom of  the  cylinder  is  a  second  valve  of  this  kind,  which  allows  the 
air  to  pass  from  below  into  the  cylinder,  but  does  not  permit  it  to 
return.  The  aperture  to  this  valve  corresponds  with  the  tube 
KLJWN;  and  at  the  extremity  .TV  of  this  tube  is  fixed  a  plate  of 
ground  glass  OP.  But  the  opening  of  the  tube  N  is  a  little  above 
this  plate,  and  has  a  stop-cock  by  which  it  may  be  at  any  time 
closed. 

In  most  experiments  we  place  on  the  plate  a  glass  receiver  ^,  the 
bottom  of  which  is  ground  with  emery  in  order  that  it  may  be  fitted 
exactly  to  the  glass  plate,  and  exclude  the  air.  It  is  sufficient  to 
press  the  receiver  a  little  when  we  place  it  upon  the  plate,  in  order 
to  make  it  adhere.  Lastly,  at  some  place  L  in  the  tube  is  a  stop- 
cock, fitting  the  tube  exactly,  so  as  to  establish  or  cut  off  at  pleasure, 
the  communication  between  the  tube  and  the  external  air. 

Sometimes,  instead  of  two  valves,  there  is  simply  a  stop-cock 
placed  in  the  tube  KL  immediately  below  the  cylinder.  This  is 
pierced  in  such  a  manner  as  to  open  at  pleasure  a  communication 
between  the  cylinder  and  the  tube,  the  cylinder  and  t{ie  external  air, 
or  fhe  external  air  and  the  tube.  This  arrangement  has  some  incon- 
veniences, but  it  has  also  many  advantages. 


142  Aeriform  Bodies. 

316.  Suppose  the  receiver  placed  on  the  plate,  the  opening  at  L 
closed,  and  the  piston  lowered  to  the  bottom  of  the  cylinder.     If  this 
piston  he  raised  a  vacuum  is  produced  below  it ;  the  air  in  the  re- 
ceiver and  tube,  therefore,  has  no  counter  pressure  above  the  valve 
K,  of  course  it  will  open  this  valve  and  expand  into  the  cylinder. 
In  this  manner  the  enclosed  mass  of  air  becomes  already  rarefied. 
If  the  piston  be  thrust  down  again,  the  air  which  had  entered  the 
cylinder  cannot  return  into  the  tube,  but  escapes  through  the  valve 
of  the  piston.     If  the  piston  be  made  to  rise  and  fall  successively,  at 
every  ascent  some  portion  of  the  air  in  the  receiver  passes  into  the 
cylinder ;  and  at  every  descent  the  air  which  had  passed  into  the 
cylinder  is  expelled  through  the  valve  of  the  piston.     Thus  the  air 
becomes  more  and  more  rarefied.     But  it  is  impossible  to  create  a 
perfect  vacuum,  because  after  a  certain  time,  we  reach  a  limit  at 
which  we  can  no  longer  produce  any  effect.     This  takes  place  when 
the  air  is  so  much  rarefied  that  its  elastic  force  is  not  sufficient  to 
raise  the  valve  K.     When  we  wish  to  introduce  the  exterior  air  into 
the  receiver,  we  turn  the  stop-cock  at  />.* 

317.  One  appendage  to  the  air-pump,  which  is  very  necessary  in 
exact  experiments,  is  a  small  syphon  barometer  called  a  gtiage.     It 
consists  of  a  recurved  tube,  ABC  (Jig-  51),  of  which  one  of  the 
branches  A  is  closed,  and  the  other  B  open.     The  space  ABF  is 
filled  with  mercury ;  and  as  the  whole  instrument  is  only  about  6  or 
7  inches  high,  the  pressure  of  the  air  in  the  open  branch  will  cause  the 
mercury  to  rise  to  the  top  of  the  other.     This  tube  is  attached  to  a 
small  support,  in  such  a  manner  that  we  can  place  it  upon  the  plate 
of  the  machine,  and  cover  it  with  the  receiver.     Between  the  two 
branches  of  the  tube  is  a  scale  DE,  which  is  divided  into  inches  and 
tenths.     When  we  place  this  apparatus  under  the  receiver,  and  be- 

*  It  is  obvious  from  this,  that  all  the  changes  Which  tend  to  render 
the  valve  more  sensible,  or  even  to  supply  its  place  altogether, 
as  may  be  done,  for  example,  by  polished  plates  sliding  upon  one 
another,  must  be  so  many  improvements  in  the  machine,  giving  it  a 
greater  power  of  exhaustion.  In  this  view  nothing  can  be  more 
complete  than  the  air-pumps  constructed  by  Fortin  at  Paris.  They 
create  a  vacuum  so  nearly  perfect,  that  the  tension  indicated  by  the 
instrument  attached  for  this  purpose  never  exceeds  what  is  inevitably 
produced  by  the  vapour  of  the  water  which  is  always  disengaged 
from  the  sides  of  the  vessels  to  which  it  adheres. 


Condensing  Pump.  143 

grn  the  exhaustion,  we  soon  perceive  that  the  air  does  not  press  the 
mercury  with  sufficient  force  to  make  it  rise  to  A.  It  therefore 
descends  in  this  tube  and  rises  in  the  other,  so  that  we  can  see  at 
each  stroke  the  altitude  of  the  column  reduced  to  an  equilibrium 
with  the  rarefied  air. 

This  instrument  indicates  properly  the  pressure  which  the  rarefied 
air  exerts  by  the  force  of  its  elasticity  ;  but  we  shall  see  in  the  fol- 
lowing chapter  that  this  is  proportional  to  its  density.  If  we  com- 
pare the  column  of  mercury  which  is  supported  by  the  pressure  of 
the  rarefied  air,  with  die  height  of  the  barometer,  we  shall  have  the 
ratio  of  the  rarefaction.  Suppose  the  barometer  to  be  at  30  inches, 
and  that  the  guage  placed  under  the  receiver  indicates  \  an  inch ; 
we  conclude  that  the  air  is  rarefied  in  the  ratio  of  \  to  30,  or  of  1  to 
60.  Those  are  considered  as  very  good  pumps,  in  which  the  mer- 
cury can  be  reduced  to  an  elevation  of  I  ^  or  2  tenths  of  an  inch. 
Ordinarily  we  are  satisfied  if  the  air,  at  its  greatest  rarefaction, 
does  not  exert  a  pressure  equivalent  to  more  than  half  an  inch  of 
mercury. 

318.  Instead   of  placing   this  guage  under  the   receiver,  it   is 
still  better  to  attach  it  to  the  pump  itself.     For  this  purpose,  the 
open  branch  of  the  tube  should  be  longer  than  the  other,  and  be  fur- 
nished with  a  stop-cock.     This  branch  should  communicate  by  some 
means  or  other  with  the  tube  LM  (Jig-  50)  in  which  the  air  is  rare- 
fied to  the  same  degree  as  in  the  receiver.  Such  a  disposition  has  the 
advantage  of  showing  the  rarefaction  in  any  vessel  whatever,  which 
may  be  substituted  in  place  of  the  receiver. 

319.  It  has  been  objected  to  the  use  of  the  guage,  that  it  does  not 
afford  any  exact  result,  because  the  humidity  always  adhering  to  the 
glass  plate  and  the  sides  of  the  receiver,  produces  elastic  vapour 
while  the  exhaustion  is  going  on ;  and  consequently,  that  what  we 
observe  is  not  the  simple  effect  of  rarefaction.     But  Laplace  has 
deduced  a  formula  from  the  experiments  of  Dalton  which  enables  us 
to  estimate  this  quantity  ;  and  Smeaton  has  invented  a  guage  against 
whicb  this  objection  cannot  be  urged. 


Condensing  Pump. 

320.  We  can  only  rarefy  the  air  by  means  of  the  apparatus  above 
described.  The  instrument  for  condensing  the  air  is  still  more  simple. 


144 


Aeriform  Bodies. 


The  piston  EF  is  without  a  valve.  It  is  sufficient  that  the  air  can 
enter  the  cylinder  by  a  small  opening  hi  the  side  immediately  below 
the  highest  point  to  which  the  piston  can  rise.  The  valve  K  must 
be  disposed  in  such  a  manner  that  the  air  can  pass  from  the  cylinder 
into  the  tube  KL,  without  being  allowed  to  return.  If  we  wish  to 
condense  the  air  in  a  receiver,  it  is  necessary  to  confine  this  receiver 
very  firmly  to  the  plate  ;  otherwise  the  force  of  the  condensed  air 
will  remove  it.  Very  strong  vessels  also  are  necessary,  because  they 
are  exposed  to  great  pressure  outward.  To  ascertain  the  degree  of 
condensation,  we  adapt  to  the  receiver  a  barometer  much  longer 
than  those  which  serve  to  measure  the  ordinary  pressure  of  the  atmo- 
sphere. 


Mechanical  Properties  of  Air. 

321.  The  mechanical  properties  of  air,  which,  like  itself,  are 
imperceptible  to  our  senses,  are  manifested  by  their  effects ;  and 
these  are  ascertained  by  means  of  the  barometer  and  air-pump. 
The  mechanical  properties  of  any  gas  may  be  reduced  to  two  j 
gravity  and  dilatability.  The  existence  and  nature  of  the  first  have 
been  already  considered.  We  proceed,  therefore,  to  treat  of  the 
dilatability  of  air. 


1.  Dilatability  of  Air. 

322.  The  dilatability  of  air  consists  in  this,  that  every  portion 
of  air  enclosed  shows  a  tendency  to  dilate  itself  and  to  occupy  a 
greater  space.  As  each  liquid  by  the  mere  force  of  its  gravity  ex- 
erts a  pressure  against  the  sides  of  the  vessel  which  contains  it,  so 
every  portion  of  air,  however  small,  by  the  mere  force  of  its  dilata- 
bility presses  all  the  sides  of  the  vessel  which  confines  it.  This  force 
is  greater  in  proportion  as  the  volume  is  more  condensed.  A  liquid 
needs  only  to  be  confined  at  the  bottom  and  sides  ;  an  aeriform  fluid 
must  be  confined  on  every  part.  The  smallest  mass  of  air  expands 
when  there  is  room,  and  fills  all  the  space  left  to  it.  Even  at  the 
state  of  the  greatest  rarefaction  which  we  can  produce,  the  air  still 
exerts  a  certain  pressure  against  the  sides  of  the  vessel  containing  it, 
which  may  be  measured  by  the  guage.  Reciprocally,  every  portion 


Dilatability  of  the  Air.  145 

of  air  may  be  compressed  into  a  less  space  than  that  which  it  occu- 
pies ;  only  its  pressure  against  the  sides  of  the  vessel  becomes 
greater,  the  more  it  is  condensed. 

No  direct  experiment  can  decide  what  are  the  limits  of  this  con- 
densation and  rarefaction,  or  whether  there  be  any.  The  law  which 
governs  the  ratios  of  the  density  and  dilatability  will  be  considered 
in  the  next  chapter.  But  the  mutual  dependence  of  the  pressure 
and  dilatability  is  self-evident.  In  the  state  of  equilibrium  they 
must  be  in  equal  ratios ;  for,  if  we  suppose  the  air  condensed  in  a 
cylinder  by  means  of  a  piston,  in  order  that  the  piston  may  be  at 
rest,  the  force  of  pressure  must  be  just  as  great  as  the  force  which 
the  dilatable  air  opposes  to  it.  Consequently,  if  the  air  is  confined 
on  all  sides  by  solid  walls,  these  walls,  according  to  the  third  law  of 
Newton,  must  resist  with  a  force  equal  to  that  which  the  dilatable  air 
exerts  against  them  ;  if  their  force  of  cohesion  is  too  weak  for  this 
they  will  burst. 

Since  the  pressure  which  a  mass  of  air  exerts  may  be  measured 
by  the  barometer,  we  have  at  the  same  time,  in  this  manner,  a  mea- 
sure of  the  dilatability. 

323.  The  dilatability  of  the  air  may  also  be  rendered  evident, 
without  the  use  of  the  air-pump,  by  means  of  very  simple  but  in- 
structive experiments,  which  I  proceed  to  describe.  Into  a  glass  flask 
AB  (Jig-  52)  having  a  long  and  narrow  neck,  we  introduce  so 
much  water  that  when  inverted  as  represented  in  the  figure,  the 
water  shall  rise  to  about  half  the  length  of  the  neck.  We  mark  this 
place  with  a  thread,  and  having  exactly  closed  the  orifice  B  of  the 
flask  AB  with  the  finger,  we  immerse  it  under  water  in  a  larger 
vessel  DEF.  If  we  immerse  it  to  the  thread,  the  water  will  be  at 
the  same  height  within  and  without  the  flask,  for  the  interior  air  is 
then  of  the  same  density  that  it  was  before  being  closed.  It  is,  there- 
fore, in  equilibrium  with  the  external  air ;  and  notwithstanding  the 
smallness  of  its  mass,  it  exerts  by  its  dilatability  the  same  pressure 
upon  the  water  contained  in  the  neck  of  the  flask,  as  the  exterior 
air,  by  its  gravity,  exerts  upon  the  water  without  the  neck.  If  we 
immerse  the  flask  AB  lower,  for  example,  so  that  the  thread  shall 
be  at^r,  the  pressure  of  the  external  air  is  added  to  that  of  the  col- 
umn of  water  CG  ;  and  the  interior  air  by  virtue  of  its  dilatability, 
which  renders  it  compressible,  must  rise  above  the  thread.  On  the 
contrary,  if  we  raise  the  flask  AB  so  that  the  thread  shall  be  at  H, 
for  example,  the  air,  by  its  dilatability,  must  expand  so  that  its 

Elem.  19 


J4&  Aeriform  Bodies. 

pressure,  added  to  that  of  the  column  of  water  in  the  neck  of  the 
flask  which  is  now  above  C,  shall  counterbalance  the  external  pres- 
sure of  the  air. 

Hence  we  see  how  easy  it  is  to  give  a  mass  of  inclosed  air  a  dila- 
tability  equal  to  the  pressure  of  the  external  air.  The  different 
gases  act  in  the  same  manner  as  atmospheric  air  under  similar  cir- 
cumstances. 

324.  By  means  of  the  air-pump  we  can  observe  the  dilatability  of 
the  air  in  more  than  one  way,  and  with  very  considerable  rarefac- 
tions and  condensations. 

(1.)  The  operation  of  rarefaction  and  condensation,  of  itself  de- 
monstrates the  dilatability  of  the  air. 

(2.)  A  bladder  apparently  exhausted  of  air,  is  distended  when  ex- 
posed in  the  vacuum  of  an  air-pump. 

(3.)  Figure  53  represents  a  Heron's  fountain.  This  consists  of 
a  closed  vessel  AB,  of  any  form,  about  half  filled  with  water,  to 
which  is  applied  a  tube  CD,  the  inferior  orifice  of  which  nearly 
touches  the  bottom,  and  the  upper  orifice  of  which  is  terminated  by 
a  pretty  sharp  point.  The  mouth  OO  of  the  vessel  is  accurately 
closed,  except  where  the  tube  passes.  If  in  such  an  apparatus  the 
air  is  first  condensed  above  the  water  by  means  of  a  bellows  or  a  con- 
densing pump,  when  we  let  it  return  to  its  natural  state,  the  water  is 
forced  through  the  opening  D,  and  jets  out  with  more  or  less  force, 
according  as  the  air  is  more  or  less  condensed.  If  we  put  such 
an  apparatus  with  the  air  not  condensed,  under  the  receiver  of  an  air- 
pump,  and  rarefy  the  external  air,  the  same  effects  are  produced. 

(4.)  The  force  of  compressed  air  is  exemplified  in  a  very  striking 
manner  in  the  air-gun. 

Pressure  of  the  Mr. 

325.  Since  the  pressure  and  dilatability  of  the  air  are  always  in 
equal  ratios,  it  is  immaterial  whether  we  say  that  the  pressure  which 
the  air  exerts  upon  a  given  surface  is  the  effect  of  the  gravity  of  the 
atmosphere,  or  that  it  is  produced  by  the  dilatability  of  the  air.    The 
air  which  surrounds  us  is  pressed  by  the  whole  weight  of  the  atmo- 
sphere ;  and  it  thus  acquires  at  each  point  a  dilatability  equal  to  the 
weight  which  compresses   it.     This  force   of  dilatation  must  be 
always  the  same  at  equal  heights.     Consequently,  at  equal  heights 
above  the  surface  of  the  earth,  the  mercury  in  the  barometer  must 


Pressure  of  the  Air.  147 

also  rise  to  the  same  degree,  whether  in  free  air  or  in  inclosed 
spaces,  provided  that  these  have  the  smallest  communication  with  the 
external  air,  and  provided  the  places  of  observation  are  not  too  re- 
mote from  each  other.  The  force  of  this  pressure  has  already  been 
determined. 

326.  By  means  of  the  air-pump  we  may  observe  this  pressure  in 
different  ways. 

(1.)  If  we  place  on  the  plate  of  the  air-pump  a  metallic  cylinder 
open  at  both  ends,  and  attach  a  bladder  to  the  upper  orifice  ;  when 
we  exhaust  the  air  from  beneath  it,  the  bladder  will  at  first  be 
strongly  pressed  and  will  at  length  break.  A  glass  plate  attached  to 
the  cylinder  by  wax  will  break  still  more  easily. 

(2.)  If  we  place  on  the  plate  a  glass  cylinder  open  at  both  ends, 
and  close  the  upper  orifice  with  a  wooden  vessel  disposed  for  the 
purpose  and  filled  with  water,  when  we  exhaust  the  air  below,  the 
water,  pressed  by  the  weight  of  the  external  air,  will  penetrate 
through  the  wood  and  fall  in  drops.  In  some  circumstances  mercury 
will  do  the  same,  and  fall  like  a  fine  shower  of  silver. 

(3.)  The  phenomenon  presented  by  the  Magdeburg  hemispheres 
is  to  be  referred  to  this  cause.  Two  hemispheres  of  metal  are  dis- 
posed in  such  a  manner  that  their  edges  are  exactly  fitted  to  each 
other.  To  the  one  is  attached  a  ring,  and  to  the  other  a  stop-cock 
and  pipe  which  may  be  screwed  to  the  air-pump.  The  edges  are 
covered  with  tallow  to  prevent  the  admission  of  air.  So  long  as  the 
internal  air  has  the  same  dilatability  as  the  external,  the  hemispheres 
are  easily  separated.  But  if  we  exhaust  the  air  from  within,  they 
are  so  strongly  pressed  together  by  the  external  air,  that  a  very  great 
force  is  required  to  separate  them.  We  may  estimate  this  force  in 
pounds  by  multiplying  by  47  the  square  of  the  diameter  of  the  sphere 
expressed  in  inches.* 

*  If  the  diameter  of  the  sphere  is  r,  the  plane  of  the  great  circle 
where  the  separation  must  take  place,  will  be  ra  n,  n  being  the  semi- 
circumference  of  the  circle  whose  radius  is  1.  If  r  is  given  in  inches, 
j»r2  is  the  surface  expressed  in  square  inches.  The  pressure  of  the 
air  upon  every  square  inch  is  about  15  Ib.  Consequently,  the  total 
pressure  is  equal  to  15  r*  n;  but  since  it  =  3,14,  15 n  =  47.  This 
result  is  always  a  little  greater  than  the  actual  pressure,  because  it  is 
impossible  to  exhaust  the  air  completely ;  but  it  is  augmented  some- 
what by  the  cohesion  of  the  hemispheres  independently  of  the  action 
of  the  air. 


148  Aeriform  Bodies. 

(4.)  The  different  kinds  of  syphon  barometers  exhibit  phenomena 
which  can  only  be  explained  by  the  pressure  of  the  air. 

Gravity  oftfo  Air. 

627.  If  the  air  is  a  heavy  fluid,  every  body  immersed  in  it  must 
lose  as  much  of  its  weight  as  is  equal  to  the  weight  of  the  fluid  whose 
place  it  occupies.  Thus,  if  we  attach  to  a  very  delicate  balance  a 
light  body  of  considerable  volume,  for  example,  a  piece  of  cork,  and 
put  it  in  equilibrium  ;  and  then  place  the  balance  under  the  receiver 
of  the  air-pump,  and  produce  a  vacuum,  the  body  will  preponderate 
because  it  loses  so  much  the  less  of  its  true  weight  as  the  surround- 
ing air  is  more  rarefied. 

By  weighing  a  body  in  the  air  we  obtain  a  result  too  small,  when 
its  density  is  less  than  that  of  the  weight  opposed  to  it ;  but  too  large 
in  the  opposite  case  ;  and  exact,  when  the  densities  of  the  two  are 
equal. 

328.  In  order  to  weigh  the  air  accurately,  we  make  use  of  a  glass 
globe  as  light  as  possible,  and  about  5  or  6  inches  in  diameter.  At 
an  opening  is  fixed  a  pipe  with  a  stock-cock,  which  may  be  screwed 
to  the  plate  of  the  air-pump.  The  cubic  capacity  of  the  globe,  after 
the  stop-cock  is  turned  so  as  to  shut  it,  must  be  determined  in  the 
most  exact  manner.  We  then  exhaust  the  air  as  perfectly  as  possi- 
ble, turn  the  stop-cock,  remove  the  globe,  and  weight  it  with  a  very 
accurate  balance.  We  then  open  the  globe  and  let  it  fill  with  air  ; 
it  thus  becomes  heavier,  and  we  ascertain  how  much  the  weight  is 
augmented.  The  excess  is  the  weight  of  the  air  contained  in  the 
globe.  If  we  divide  this  weigh  by  the  capacity  of  the  globe,  ex- 
pressed in  cubic  inches,  we  shall  have  the  weigh  of  a  cubic  inch  of 
air.  If  the  experiment  is  to  be  performed  with  extreme  precision, 
we  make  use  of  a  guage  to  ascertain  how  many  cubic  inches  of  air 
remain  in  the  globe  in. order  that  we  may  deduct  them  from  its 
cubic  capacity. 

Moreover,  as  the  weight  of  the  air  varies  with  the  state  of  the 
barometer  and  thermometer,  the  experiment  must  be  performed  at 
a  determinate  state  of  these  two  instruments,  or  a  reduction  must  be 
made  by  calculation.  We  ordinarily  select  for  this  purpose,  the 
temperature  of  60°,  and  a  barometric  height  of  30  inches.  But  it 
is  better  to  have  recourse  to  reductions,  for  it  is  almost  impossible  to 
combine  these  two  conditions  exactly. 


Equilibrium  of  ,#tr,  149 


Specific  Gravity  of  other  Gases. 

329.  After  the  globe  is  weighed,  if  we  introduce  some  other  gas, 
we  can  find  the  weight  of  it  in  the  manner  indicated  for  air.     We 
can  thus  ascertain  the  weight  of  a  cubic  inch,  and  this,  as  before  ob- 
served, is  the  usual  method  of  expressing  the  specific  gravity  of  the 
gases. 

330.  We  shall  give  the  specific  gravity  of  some  of  the  gases,  as 
determined  by  the  most  careful  experiments,  made  at  a  barometric 
height  of  30  inches,  and  at  the  temperature  of  60°  of  Fahrenheit. 

Water  being  1.  At.  Air  being  1. 

Atmospheric  ail-  0,00122  1,0000 

Azote  0,00119  0,9722 

Oxygen  0,00136  1,1111 

Carbonic  acid  0,00186  1,5277 


CHAPTER  XXX. 

Equilibrium  of  Mr,  or  First  Principles  of  Aerostatics. 

331.  WHILE  the  air  is  considered  as  a  heavy  fluid  the  essential 
laws  of  hydrostatics  must  be  applied  to  it. 

( 1 .)  Every  pressure  made  in  air  propagates  itself  in  all  directions 
in  the  same  manner  as  in  liquids. 

(2.)  In  a  state  of  equilibrium  the  pressure  must  be  equal  upon  all 
the  points  of  every  horizontal  plane ;  but,  on  account  of  the  great 
levity  of  the  air,  this  pressure  must  diminish  as  we  ascend  much 
more  slowly  than  in  liquids.  But  the  law  of  this  diminution  is  not 
the  same  for  air  and  liquids,  as  we  shall  see  hereafter. 

(3.)  The  pressure  of  the  air  upon  a  given  surface  may  be  deter- 
mined in  the  same  manner  as  that  of  water ;  that  is,  it  is  equal  to 
the  weight  of  a  prism  of  mercury  whose  base  is  the  surface  pressed, 
and  whose  altitude  is  the  height  of  the  mercury  in  the  barometer. 
But  on  account  of  the  slow  diminution  of  the  pressure  of  the  air,  we 
do  not  perceive  any  difference  whether  the  plane  be  horizontal, 
vertical,  or  oblique,  unless  it  be  one  of  great  magnitude.  Neither 


150  Aeriform  Bodies. 

does  it  make  any  difference  whether  the  pressure  proceeds  from  free 
air  or  from  air  inclosed,  provided  the  latter  have  the  same  dilata- 
bility  as  the  external  air.  We  have  already  explained  the  method 
of  estimating  this  pressure. 

(4.)  Any  body  immersed  in  the  air  loses  a  portion  of  its  weight 
equal  to  that  of  the  air  displaced  by  it,  as  we  have  seen  in  the  cas« 
of  liquids. 

(5.)  A  body  which  is  lighter  than  the  same  volume  of  atmospheric 
air,  rises  until  it  is  in  equilibrium  with  the  surrounding  fluid,  which,  as 
we  shall  see  hereafter,  always  becomes  rarer  as  we  ascend.  On 
this  is  founded  the  theory  of  balloons ;  as  well  those  in  which  the 
air  is  rarefied,  according  to  the  method  of  Montgolfier,  as  those 
which  are  filled  with  hydrogen  gas,  according  to  the  method  of  M. 
Charles. 


Law  of  Mariotte  ;  or  Ratio  of  the  Pressure  and  Elasticity  to  the 
Density  or  Specific  Gravity. 

332.  The  effects  of  dilatability  constitute  a  fundamental  differ- 
ence between  elastic  fluids  and  liquids.  Among  these  effects  we 
notice  particularly  the  diminution  of  density  depending  upon  the 
height.  The  inferior  strata  of  the  air  are  pressed  by  the  whole 
weight  of  the  atmosphere.  In  the  higher  regions  the  weight  be- 
comes more  and  more  feeble,  and  consequently  the  density  of  the 
air  diminishes.  But  in  order  to  determine  the  law  according  to 
which  the  density  diminishes,  we  must  first  determine  by  experiment 
what  is  in  general  the  ratio  of  the  pressure  and  density  in  a  mass  of 
air. 

333.  The  following  law  which  is  extremely  simple  is  abundantly 
confirmed  by  experiment. 

The  density  of  a  mass  of  air  inci'eases  and  decreases  in  the  same 
ratio  as  the  pressure,  unless  some  change  takes  place  in  the  tempera- 
ture, or  in  the  chemical  combination  of  the  parts. 

As  the  pressure  and  dilatability  are  always  equal,  and  as  the  den- 
sity and  specific  gravity  are  synonymous  terras,  the  following  is  only 
a  different  enunciation  of  the  same  law. 

The  dilatability  of  a  mass  of  air  is  proportional  to  its  specific 
gravity,  as  long  as  its  temperature  and  chemical  combination  remain 
the  same. 


Law  ofMariotte.  151 

This  important  principle  of  aerostatics  is  called  the  law  of  Ma- 
riotte,  although  it  was  discovered  in  England  by  Sir  Robert  Boyle 
and  his  disciple,  Townley,  a  short  time  before  Mariotte  recognised 
it  in  France.  The  experiments  which  serve  to  show  the  exactness 
of  this  law,  are  briefly  as  follows  ; 

In  order  to  measure  the  condensation  of  air  by  pressure,  we  em- 
ploy a  tube  of  glass  that  is  recurved  like  a  syphon  barometer  (Jig.  49), 
with  this  difference,  that  the  short  branch  is  closed  at  G,  and  the 
long  branch  open  at  A.  It  is  also  convenient  to  give  this  last  a 
length  of  several  feet.  The  air  enclosed  between  G  and  CD  will 
be  compressed  at  the  same  time  by  the  column  of  mercury  EF,  and 
by  the  external  air,  since  A  is  open.  This  last  pressure  is  equal  to 
that  of  the  barometric  column.  If  then  we  gradually  fill  the  long 
branch  with  mercury,  and  always  measure  the  space  which  the  en- 
closed air  occupies,  it  will  be  easily  seen  how  the  pressure  and  den- 
sity may  be  compared  ;  for  the  density  is  in  the  inverse  ratio  of  the 
space  CG,  occupied  by  the  air.  In  order  to  measure  the  rarefac- 
tion of  the  air  produced  by  a  diminution  of  pressure,  we  employ  a 
straight  barometric  tube,  open  at  its  lower  extremity,  and  provided 
with  a  stop-cock  at  the  other.  This  stop-cock  being  open,  we  im- 
merse the  lower  end  of  the  tube  in  a  vessel  filled  with  mercury, 
until  there  remains  only  an  inch  or  two  of  air  in  the  tube  ;  we  then 
turn  the  stop-cock,  and  gradually  raise  the  tube.  In  proportion  as 
we  raise  it,  the  enclosed  air  dilates  ;  but  a  column  of  mercury  rises 
under  it,  above  the  surface  of  the  exterior  mercury.  We  measure 
from  time  to  time,  the  space  occupied  by  the  enclosed  air,  and  the 
height  of  the  mercurial  column  in  the  tube.  The  force  with  which 
the  enclosed  air  is  pressed  is  always  equal  to  that  of  the  barometric 
column,  minus  the  column  which  rises  in  the  tube.  The  pressure 
and  density  may  thus  be  compared  under  these  circumstances,  as  in 
the  preceding  experiment. 

In  order  to  perform  experiments  of  this  kind  upon  a  small  scale, 
we  employ  a  pretty  long  barometric  tube  AB  (fig  54),  which  is 
open  at  A  and  closed  at  B,  and  whose  interior  diameter  is  about 
|  line.  This  tube  must  be  of  a  uniform  size,  especially  from  A  to 
half  its  length.  To  this  is  added  a  scale  divided  into  inches  and 
parts.  Into  this  tube  we  introduce  a  column  of  mercury  of  about  5 
or  6  inches  in  length,  which  is  done  by  expelling  a  portion  of  the 
air  by  heat.  Let  us  suppose  that  the  column  of  mercury  is  nearly 
in  the  middle  of  the  tube.  If  we  hold  this  in  a  vertical  position, 


152  Aeriform  Bodies. 

with  the  open  extremity  A  upward,  the  pressure  supported  by  the 
enclosed  air  BD,  will  be  equal  to  the  pressure  of  the  column  of 
mercury  CD,  added  to  that  of  the  barometric  column.  If,  on  the 
contrary,  we  invert  the  tube,  the  open  extremity  being  downward, 
the  pressure  supported  by  the  enclosed  air  becomes  equal  to  that  of 
the  barometric  column,  minus  the  column  CD.  In  the  horizontal 
position  it  is  just  equal  to  that  of  the  barometric  column.  We  can, 
therefore,  in  these  three  cases,  compare  the  pressure  with  the  space 
occupied  by  the  enclosed  air.  Such  an  instrument  resembles  in  its 
essential  parts  the  manometer  of  Varignon  or  of  Wolf. 

334.  The  importance  of  the  law  of  Mariotte  requires  that  we 
should  know  exactly  what  are  its  limits  and  conditions. 

(1.)  For  atmospheric  air  it  has  been  found  to  be  exact  to  an  oc- 
tuple condensation,  and  beyond  a  centuple  rarefaction.  We  cannot 
decide  whether  it  is  exact  for  all  imaginable  condensations  and  rare- 
factions. The  advocates  for  the  atomic  theory  must  deny  it.  On 
the  contrary,  those  who  maintain  the  dynamic  theory,  must  support 
the  affirmative.  So  far  as  practice  is  concerned,  it  is  sufficient  to 
know  that  this  law  is  applicable  to  all  experiments  we  have  occasion 
to  make. 

(2.)  Experiments  have  been  made  only  at  mean  temperatures. 
But  it  is  a  necessary  consequence  of  the  experiments  of  Gay-Lussac 
and  Dalton,  that  the  same  law  should  obtain  for  all  temperatures  ; 
for  if  in  the  two  masses  of  air  A  and  B,  the  chemical  properties  and 
mean  temperatures  of  which  are  the  same,  the  density  is  proportional 
to  the  pressure,  the  reason  of  this  proportionality  must  remain  the 
same  for  all  temperatures,  since  the  masses  of  air  are  dilated  equally 
by  heat. 

(3.)  These  experiments  upon  pressure  were  made  only  with 
atmospheric  air  ;  and  it  slill  remains  to  determine  by  immediate  ex- 
periment, whether  the  law  of  Mariotte  is  exact  for  all  other  elastic 
fluids.  Nevertheless,  since  in  the  experiments  upon  dilatation,  which 
we  have  before  considered,  heat  acts  uniformly  upon  all  these 
fluids,  it  is  extremely  probable  that  mechanical  pressure  also  acts 
uniformly  upon  all.  Another  fact  which  renders  this  opinion  ex- 
tremely probable,  is,  that  all  the  experiments  made  with  atmospheric 
air,  have  always  given  the  same  results,  although  the  air  employed 
in  the  different  experiments,  may  have  differed  in  its  chemical  com- 
position. 


Law  of  the  Decrease  of  Density  in  the  Atmosphere.        153 

For  these  different  reasons,  until  the  question  is  perfectly  settled, 
>ve  may  admit,  as  a  very  probable  hypothesis,  that  the  law  of  Ma- 
riotte  is  applicable  to  all  elastic  fluids. 

(4.)  Hitherto  we  have  considered  the  application  of  the  law  only 
with  respect  to  an  elastic  fluid  insulated  from  others.  But  it  may 
be  asked,  whether  the  density  of  two  masses  of  air,  A.  and  B,  the 
chemical  natures  of  which  are  different,  is  proportional  to  the  pressure 
indicated  by  their  volumes.  This  question  must  be  answered  in  the 
negative  ;  for  experiment  shows  that  different  masses  of  air,  at  the 
same  pressure  and  temperature,  have  different  specific  gravities. 
They  require  therefore  different  pressures  to  give  them  die  same 
density. 


Law  by  which  the  Density  of  the  Air  decreases  as  we  ascend  into  the 
Atmosphere. 

336.  By  certain  simple  mathematical  operations,  we  can  deduce 
from  the  law  of  Mariotte,  the  principal  theorem  of  aerostatics,  which 
is  the  following ; 

In  a  state  of  equilibrium  the  density  of  the  air  must  decrease,  as  we 
ascend,  in  a  geometrical  ratio,  when  the  chemical  nature  and  tempera- 
ture of  the  column  are  the  same  throughout  the  whole  extent. 

Thus,  if  we  divide  the  column  of  air  ABCD,  (Jig.  55)  into  strata 
of  a  thickness  taken  at  pleasure,  but  all  equal  to  each  other,  AEFB, 
EGHF,  &c.,  the  density  of  the  air  decreases  in  a  geometrical  series 
at  the  points  E,  G,  I,  L,  that  is,  each  succeeding  one  is  less  than 
the  preceding  in  the  same  ratio.* 


*  Let  us  suppose  that  the  strata  of  air  are  taken  so  thin,  that  the 
density  of  each  may  be  considered  as  uniform  throughout.  Let  the 
density  of  the  inferior  stratum  be  A,  that  of  the  next  B,  of  the  third 
C,  and  so  on.  Moreover,  let  a  be  the  weight  of  the  whole  column 
of  air  AECD  ;  b  its  weight  when  the  inferior  stratum  is  taken  away, 
c  its  weight  when  the  second  is  taken  away,  and  so  on.  Then  the 
weight  of  the  first  stratum  =.  a  —  b,  that  of  the  second  =6  —  ct 
that  £»f  the  third  =.  c  —  d,  and  so  on.  Now  the  density  of  two 
bodies  of  the  same  volume  is  in  general  as  their  weights.  Conse- 
quently A:E::a  —  b:b  —  c.  But,  according  to  the  law  of  Ma- 
riotte, the  density  of  two  masses  of  air  is  proportional  to  the  pressure 

Elem.  20 


154  Aeriform  Bodies. 

336.  This  theorem  may  be  expressed  in  various  ways ;  particu- 
larly in  the  six  following.     When  a  column  of  air  has  throughout 
the  same  temperature  and  the  same  chemical  nature,  we  may  con- 
sider as  decreasing  in  a  geometrical  series , 

(1.)  The  density  of  the  air. 

(2.)  Its  specific  gravity. 

(3.)  The  weight  of  the  incumbent  mass. 

(4.)  The  pressure  which  the  air  suffers  and  exerts. 

(5.)  The  elasticity  of  the  air. 

(6.)  The  barometric  height. 

Nos.  1,  2,  are  different  expressions  for  the  same  thing.  Nbs.  3. 
4,  5,  6,  are  also  different  ways  of  considering  what  is  in  itself  the 
same  ;  for  the  weight  of  the  air  above  is  only  the  pressure  supported 
by  the  air  below,  or  exerted  by  this  same  air,  in  a  state  of  equilibri- 
um. Moreover,  the  pressure  which  a  mass  of  air  exerts  is  equal  to 
its  elasticity,  and  the  barometric  height  is  the  measure  of  the  pressure. 

Thus,  according  to  the  law  of  Mariotte,  nos.  1,2,  on  the  one 
hand,  and  nos.  3,  4,  5,  6,  on  the  other,  indicate  properties  mutually 
dependent ;  and  consequently  the  general  sense  of  the  theorem  is 
this  ;  admitting  the  law  of  Mariotte,  the  properties  indicated  in  1 ,  2, 
decrease  in  the  same  ratio  as  those  indicated  in  3,  4,  5,  6,  and 
vice  versa. 

337.  As  the  condition  of  this  theorem  is  thnt  the  column  of  air 
shall  have  throughout  the  same  temperature  and  the  same  chemical 
nature,  we  must  not  expect,  in  reality,  to  find  this  geometrical  de- 

which  they  support.  Consequently,  A  :  B  :  :  b  :  c.  Omitting  the 
common  ratio,  we  have  a  —  b  :  b  —  c::6:c;  which  gives 

ac  —  6  c  =.  63  —  6c, 

or  a  c  =r  b  ;  whence  a  :  b  : :  b  :  c.  In  the  same  manner  we  obtain 
b  :  c  :  :  c  :  d,  also  c:d:-.d'.e,  and  so  on.  The  weights  a,  6,  c,  d, 
e,  &c.,  form,  therefore,  a  geometrical  series.  But  as  the  densities 
are  proportional  to  these  weights,  they  also  form  a  geometrical  series. 
This  demonstration  is  rigorously  true  only  for  strata  which  are  infi- 
nitely thin.  But  it  is  a  property  of  all  geometrical  series,  that  if  we 
take  out  some  of  the  intermediate  members,  these  members  form  a 
new  geometrical  series,  provided  there  is  always  an  equal  number 
of  terms  between  those  taken  out.  Hence  it  is  evident  that  this  prin- 
ciple is  correct,  when  the  altitudes  AE,  CO,  GJ,  IL,  &c.,  are  of  a 
finite  magnitude. 


Measurement  of  Heights  by  the  Barometer.  155 

crease  of  density  perfectly  exact.  But  yet  we  must  not  consider 
the  law  of  Mariotte  and  the  theorem  deduced  from  it  as  mere  hy- 
potheses to  be  admitted  or  rejected  at  pleasure.  We  have  just  as 
much  reason  to  consider  the  law  of  the  descent  of  heavy  bodies,  as 
a  mere  hypothesis,  since  the  resistance  of  the  air  occasions  devia- 
tions from  it.  All  the  motions  which  take  place  in  the  atmosphere 
are  only  continual  efforts  which  nature  makes  to  restore  the  equili- 
brium, which  secondary  causes  are  continually  disturbing.  There 
must,  therefore,  be  in  fact  a  continual  tendency  towards  this  equili- 
brium. Philosophers  cannot  reject  general  laws,  or  change  them  at 
pleasure  ;  but  they  should  endeavour  to  ascertain,  as  far  as  possible, 
the  influence  of  disturbing  forces. 


More  exact  Estimate  of  the  Influence  which  Heat  has  upon  the  Me- 
chanical Properties  of  a  Dilatable  Fluid. 

338.  We  have  akeady  mentioned  the  important  discovery  made  at 
the  same  time  by  Dalton  and  Gay-Lussac  ;  that  all  elastic  fluids  are 
equally  dilated  by  heat  when  the  pressure  is  the  same.     This  dilata- 
tion between  the  freezing  and  boiling  point,  amounts  to  0,375   or 
f  of  the  volume  which  the  mass  had  at  the  first  temperature.     This 
remarkable  discovery  enables  us  to  determine  with  great  exactness 
the  influence  which  heat  has  upon  the  density  and  elasticity  of  a 
mass  of  air. 

339.  Since  the  law  of  Mariotte  is  applicable  to  each  mass  of  elas- 
stic  fluid  ;  it  follows  reciprocally, 

That  in  a  mass  of  air  perfectly  confined  and  incapable  of  chang- 
ing its  volume,  the  elasticity  must  increase  by  being  heated  in  the 
same  ratio  as  its  volume  would  increase,  if,  the  pressure  being  the 
same,  it  had  the  power  of  dilatation. 

Hence  from  the  freezing  to  the  boiling  point  the  elasticity  of  a 
mass  of  air,  perfectly  confined,  must  increase  in  the  ratio  of 
1000  :  1375  or  8  :  11. 

340.  If  we  now  divide  the  fundamental  distance  of  Lambert's 
air  thermometer  into  375  parts;  placing  1000  at  the  freezing  point, 
and  1375  at  the  boiling  point;  and  if  we  estimate  the  temperature 
by  the  degrees  of  this  thermometer,  the  comparison  of  two  numbers 
of  this  scale  will  indicate  exactly  what,  under  these  two  tempera- 
tures, would  be  the  ratios  of  dilatation  of  a  mass  of  air,  the  pressure 


156  Aeriform  Bodies. 

remaining  the  same,  or  those  of  its  disability,  the  volume  remaining 
the  same. 

If  we  prefer  to  divide  the  fundamental  distance  into  180  parts 
instead  of  375,  conformably  to  Fahrenheit's  thermometer,  the  num- 
ber standing  at  the  freezing  point  must  be  480  (a  fourth  proportional 
to  375,  1000,  and  180,)  and  consequently  that  at  the  boiling  point, 
must  be  660.  The  numbers  of  this  scale  would  also  indicate  imme- 
diately the  ratios  of  which  we  have  spoken. 

341.  The  march  "of  an  air  thermometer  divided  into  180  parts, 
and  that  of  a  mercurial  thermometer  with  the  same  divisions,  do  not 
perhaps  perfectly  conform  to  each  other  ;  but  according  to  the  ob- 
servations of  Lambert,  they  differ  very  little  between  the  freezing 
and  boiling  points.*  If  then  we  put  480°  in  place  of  32°,  and  660° 
in  place  of  180,  or  what  amounts  to  the  same  thing,  if  we  add  480 
to  the  temperature  indicated  by  Fahrenheit's  thermometer,  these 
numbers  will  express,  in  an  approximate  manner,  the  ratios  determin- 
ed in  the  preceding  article.  Accordingly  it  might  be  useful  in 
aerostatics  and  perhaps  in  all  that  relates  to  thermometry,  to  intro- 
duce universally  the  use  of  the  air  thermometer  instead  of  the  mer- 
curial, or  at  least  to  determine  with  precision  the  ratios  of  the  two 
scales.f 


Measurement  of  Heights  by  the  Barometer. 

342.  The  method  of  measuring  heights  by  means  of  the  barome- 
ter, is  founded  upon  the  preceding  theorem.  This  method  is  of 
such  extensive  use  that  we  cannot  omit  giving  a  brief  account  of  it. 

[When  the  altitudes  above  the  surface  are  taken  in  arithmetical 
progression,  the  corresponding  densities,  and  consequently  the  in- 
cumbent weights  of  the  atmosphere  at  these  heights,  form,  as  we 
have  seen,  a  geometrical  series ;  in  other  words,  the  heights  are  the 
logarithms  of  the  corresponding  weights  of  the  atmosphere,  accord- 
ing to  a  particular  base,  which  may  be  determined  by  experiment. 

*  Gay  Lussac  has  recently  proved,  by  unquestionable  experiments, 
that  they  are  rigorously  the  same  when  the  air  and  tubes  are  per- 
fectly dried. 

t  This  has  since  been  done  by  Gay-Lussac,  and  his  results  have 
been  confirmed  by  MM.  Petit  and  Dulong. 


Measurement  of  Heights  by  the  Barometer.  157 

Distinguishing  these  logarithms  by  A,  if  we  denote  any  two  heights 
by  h,  h',  and  the  corresponding  weights  of  the  atmosphere,  as  deter- 
mined by  the  barometer,  by  w,  «/,  we  shall  have 

h'  —  h  =  l  w  —  l  W1, 
or  putting  h  =  0,  h'  =  A  w  —  i  u/, 

that  is,  the  difference  of  level,  or  height  of  one  of  the  places  in  ques- 
tion above  the  other,  is  expressed  by  the  difference  of  the  logarithms 
of  the  mercurial  columns,  these  logarithms  being  constructed  upon 
a  particular  base  adapted  to  this  purpose.  Now,  since  logarithms 
are  changed  from  one  system  to  another  by  a  constant  multiplier,  we 
shall  have 

h'  =  x  (log.  w  —  log.  w), 

log.  denoting  the  common  logarithm  of  the  quantity  before  which  it 
is  placed.  Hence,  by  taking  an  object  whose  elevation  has  been 
previously  ascertained  by  other  methods,  we  readily  find,  once  for 
all,  the  value  of  the  multiplier  a?,  thus 


~~  log.  w  —  log.  w'' 

Taking  the  mean  of  a  great  number  of  observations,  conducted 
with  the  greatest  care  by  M.  Raymond,  we  find  x  equal  to  18336 
metres,  or  60156  feet,  or  10026  English  fathoms.  This  is  on  the 
supposition  of  a  temperature  of  32°.  Now  air,  by  having  its  tem- 
perature raised  from  32°  to  212°,  dilates  0,375,  or 

=  ?  ._  =  0,00223 


for  each  degree  of  Fahrenheit  ;  and,  according  to  the  law  of  Mariotte, 
if  this  air  be  reduced  back  again  to  the  same  bulk,  it  will  have  its 
pressure  increased  in  the  same  proportion.  We  can,  therefore, 
change  the  constant  multiplier  10026  to  10000,  by  supposing  the 
temperature  somewhat  lower.  Thus,  0,00223  :  0,0026  :  :  1°  :  1°,16. 
If,  therefore,  we  subtract  1°,16  from  32°,  we  shall  have  30°,84,  or 
31°  nearly,  for  the  temperature  at  which  the  constant  coefficient  is 
10000  fathoms. 

34  J.  Given  the  height  of  the  mercury  in  the  barometer  at  the 
bottom  of  a  mountain  =  29,37  inches,  and  at  its  summit  =  26,59 
inches,  to  find  the  altitude  of  the  mountain,  the  mean  temperature  at 
the  two  stations  being  26°. 


j58  Aeriform  Bodies. 

29,37     ....     log  .....     1,46700 
26,59     ....     log  .....     1,42472 

diff.     .     .     .     .     0,04318 

Approximate  height  .....     =431,8  fathoms, 
Correction  5  X  0,00223  X  431,8  =  —  4,8 

True  height    .     .     .    '     .     ,.     ,     427,0  =  2562,0  feet. 

344.  We  have  supposed  in  the  above  examples  that  the  tem- 
perature of  the  mercurial  columns  at  the  two  stations  is  the  same. 
Where  the  difference  is  considerable,  the  result  will  evidently  be 
affected  by  it.  If  the  upper  station,  for  instance,  be  the  coldest, 
which  most  frequently  happens,  the  mercurial  column  will  be  too 
short,  and  will  consequently  indicate  too  great  a  height.  The 
contraction  being  about  10000th  part  for  each  degree  of  cold,  or 
0,0025  in.  in  a  column  of  25  inches,  it  would  require  4°  differ- 
ence of  temperature  to  produce  an  effect  amounting  to  one  division 
on  the  scale  of  a  common  barometer,  where  the  graduation  is  to 
hundredths  of  an  inch. 

This  correction  is  combined  with  the  foregoing  rule  in  the  follow- 
ing formula,  in  which  t,  /',  represent  the  temperature  of  the  air,  q, 
rf>  that  of  the  mercury,  at  the  two  stations  respectively, 


A- 


Height  of  the  Atmosphere. 

345.  If  the  law  of  Mariotte  is  exact  for  all  imaginable  degrees  of 
rarefaction  and  condensation,  it  follows  from  the  fundamental  theo- 
rem, that  the  dilatation  of  the  atmosphere  is  unlimited,  since  in  a 
decreasing  geometrical  progression,  the  terms,  however  far  continued 
can  never  become  strictly  nothing.  This  statement  in  itself  contains 
no  contradiction.  But  yet  it  does  not  appear  to  accord  with  the  ob- 
servations of  astronomer?,  who  suppose  the  planets  to  move  in  an 
unresisting  medium.  We  cannot,  therefore,  determine  absolutely 
what  is  the  height  of  the  atmosphere.  It  is  however  demonstrated 
by  the  theory  of  barometric  heights,  that  at  the  height  of  40,000, 
about  45  miles,  the  air  must  be  at  least  as  rare  as  in  the  vacuum  of 
our  best  air-pumps.  Hence  we  usually  say  the  atmosphere  extends 


Motion*  of  Elastic  Fluids.  1 59 

45  miles  high.  Yet  if  we  estimate  exactly  the  height  of  some 
meteors,  such  as  the  aurora  borealis,  fire  balls,  Sic.,  we  are  com- 
pelled to  admit  that  at  the  height  of  more  than  60  miles,  there  must 
not  only  be  atmospheric  air,  but  also  many  other  substances  which 
we  should  not  expect  to  find  at  such  an  elevation. 


CHAPTER  XXXI. 

Motions  of  Elastic  Fluids. 

346.  In  considering  the  motions  of  elastic  fluids,  the  inquirer  may- 
confine  himself  almost  entirely  to  the  examination  of  atmospheric  air. 
For  the  other  gases  are  found  in  such  small  quantities,  and  fill  such 
small  spaces,  that  their  motions  have  rarely  any  peculiar  interest. 
But,  on  the  contrary,  the  immense  extent  and  great  agitations  of  the 
air  which  envelopes  the  globe,  the  continual  winds,  periodical  or 
accidental,  by  which  it  is  agitated,  are  not  only  remarkable  phe- 
nomena in  themselves,  but  in  many  respects  of  the  highest  import- 
ance to  the  interests  of  society  ;  and  we  can  judge  of  what  vast  util- 
ity the  knowledge  of  the  laws  of  these  motions  would  be,  if  we  could 
thereby  be  able  to  predict  them.     There  are  also  many  artificial 
motions  of  the  air,  the  observation  of  which  is  important.     In  many 
pneumatic  or  hydraulico-pneumatic  machines,  the  elasticity  of  the 
air  or  steam  is  the  sole  cause  of  motion.     It  is  by  the  knowledge 
of  these  motions,  that  currents  of  air  have  been  created  in  mines, 
to  drive  off  the  unwholesome  gases,  and  so   managed    in  chim- 

'nies  as  to  defend  us  from  the  inconvenience  of  smoke. 

347.  The  philosophical  mode  of  treating  these  motions,  is  first  to 
develope  their  fundamental  principles,  and  then  to  confirm  them  by 
experiment.     But  in  treating  what  is  called  pneumatics,  we  have 
not  only  to  contend  with  all  the  difficulties  which  occur  in  hydrau- 
lics, but  it  is  necessary  also  to  have  regard  to  two  peculiar  and  very 
active  causes,  which  continually  embarrass  us  in  our  researches. 
These  are  dilatability  and  heat.     Besides,  this  branch  of  physical 
science  is  entirely  without  fundamental  principles  demonstrated  by 
rigorous  reasoning,  and  confirmed  by  experiment.     For  these  rea- 
sons we  must  content  ourselves  with  general  remarks  on  the  subject 
before  us. 


1(50  Aeriform  Bodies, 

348.  We  should  be  able  to  explain  with  sufficient  accuracy  all 
the  causes  of  the  motions  which  take  place  in  the  atmosphere,  if  two 
conditions  essential  to  exact  researches,  were  not  in  most  cases  want- 
ing.    The  first  is  the  accurate  measure  of  the  effect  produced  ;  the 
second  is  a  knowledge  of  all  the  different  circumstances  which  influ- 
ence a  single  phenomenon.     We  know,  for  example,  the  general 
causes  of  wind  ;  but  we  never  know,  or  very  rarely,  of  what  force 
wind  is  the  effect,  and  to  what  distance  it  extends ;  or  what  are  the 
particular  causes  of  the  wind  which  blows  at  any  particular  time. 

349.  Instead  of  the  fundamental  laws  of  pneumatics,  we  can  only 
state  the  following  general  principle,  which  is  too  evident  to  need 
any  proof. 

Every  cause  which  acts  upon  a  mass  of  air  in  opposition  to  one  of 
the  laws  of  equilibrium)  must  produce  motion. 

As  we  have  already  stated  the  conditions  of  equilibrium,  it  will  be 
very  easy  to  form  a  clear  idea  of  the  causes  of  motion  in  the  air. 

350.  One  of  the  principal  is  heat.     The  motions  which  proceed 
from  this  cause,  though  exceedingly  varied,  are  all  produced  in  the 
same  manner.   Heat  increases  the  elasticity  of  the  air  ;  accordingly, 
when  in  any  region  of  the  atmosphere  a  mass  of  air  is  heated  more 
than  that  which  surrounds  it,  it  dilates  and  repels  in  every  direction 
the  air  which  is  colder  than  itself.     In  this  manner  the  equilibrium 
is  destroyed  ;  and  the  heated  air  becoming  lighter,  must  rise  accord- 
ing to  the  laws  of  hydrostatics ;  for  the  surrounding  air  being  colder, 
is  for  that  reason  heavier.     Reciprocally,  the  cold  air  must  descend 
and  press  towards  the  place  where  the  heat  acts  ;  the  air  then  accu- 
mulates above  the  heated  place,  which  necessarily  produces  a  cur- 
rent of  air,  propagating  itself  in  every  direction.     Heat,  therefore, 
always  produces  a  double  current  of  air,  namely,  one  tending  toward 
the  place  from  below,  and  one  in  the  opposite  direction  from  above. 
Cold  evidently  acts  in  precisely  the  opposite  manner. 

351.  According  to  this  general  theory,  the  motions  caused  by 
heat  and  cold  may  be  easily  explained,  regard  being  had  to  the  dif- 
ferent circumstances  which  modify  each  given  case. 

Thus  most  of  the  winds  proceed  from  the  heating  and  cooling  of 
different  regions  of  the  atmosphere,  particularly  the  constant  and 
periodical  winds  observed  in  the  torrid  zone. 

It  is  upon  the  same  principles  that  currents  of  air  act  in  chimnies} 
Argand  lamps,  &c.  We  may  observe,  by  means  of  a  lighted  candle, 
the  two  currents  of  air  above  mentioned,  at  the  opening  of  a  door  of 


Motions  of  Elastic  Fluids.  161 

a  heated  room.  A  current  of  air  is  also  produced  in  mines,  by 
means  of  a  well  or  a  gallery  constructed  in  them,  since  the  tempera- 
ture of  mines  is  very  different  from  the  exterior  temperature.  If 
this  difference  does  not  produce  a  sufficient  effect,  a  fire  may  be 
made  in  the  mine. 

352.  Since  at  the  same  temperature  and  pressure  every  elastic 
fluid  possesses  a  particular  degree  of  density,  everj  change  which 
takes  place  in  the  chemical  combination  of  a  mass  of  air  is,  like  beat 
and  cold,  a  cause  of  motion.     Every  incre^e  of  density  acts  like 
heat,  every  diminution  like  cold.     Now  cffemges  of  this  kind  are 
continually  taking  place  in  the  atmosphere  ;    since,  by  means  of 
organic  and  chemical  processes,  many  of  which  are  probably  un- 
known to  us,  the  air  sometimes  parts  with  one  or  more  of  its  con- 
stituent principles  to  other  bodies,   and  sometimes  combines   with 
some  of  theirs.     Here  then  must  be  an  incessant  cause  of  motions, 
which,  however,  are  rarely  violent,  because  these  changes  never  take 
place  rapidly. 

353.  Such  changes  produce  motions,  not  only  because  they  alter 
the  elastic  force  of  the  air,  but  also  because  they  increase  or  diminish 
its  mass.     When  the  mass  of  air  is  increased,  currents  are  formed 
tending  from  the  place  in  every  direction ;  when  it  is  diminished 
the  contrary  effect  is  produced.     The  most  active  cause  of  this  na- 
ture is  undoubtedly  evaporation.     It  is  found  that  there  annually 
evaporates  a  stratum  of  water  of  about  30  inches  in  thickness,  in  the 
temperate  regions  of  Europe ;  and  this  takes  place  at  the  rate  of 
about  half  an  inch  in  the  coldest  months,  and  4  or  5  inches  in  the 
warmest.  We  may  hence  imagine  how  much  the  mass  of  air  incum- 
bent upon  the  immense  surface  of  seas  and  oceans  continually  aug- 
ments, especially  in  the  torrid  zone  ;  and  we  may  attribute  to  this 
augmentation  a  part  of  the  motions  which  take  place  in  the  whole 
atmosphere. 

But  much  more  violent  motions  must  arise  from  the  opposite 
cause ;  that  is,  when  this  vapour  is  condensed  and  falls  upon  the 
earth  in  rain,  snow,  and  hail ;  these  are  particularly  remarkable  in 
violent  showers,  when  a  limited  portion  of  the  atmosphere  in  the 
course  of  a  few  hours,  parts  with  several  thousand  tons  from  its  mass  ; 
which  must  evidently  produce  currents  tending  towards  the  place 
from  every  quarter.  Indeed  this  is  what  is  often  perceived,  when 
the  course  of  a  shower  is  attentively  observed. 

Elem.  21 


162  Aeriform  Sadies. 

354.  The  motions  also  of  other  bodies,  and  particularly  of  water, 
are  communicated  to  the  air.     When  the  air  is  tranquil,  we  observe 
over  the  surface  of  each  river,  the  course  of  which  is  considerably 
rapid,  a  current  of  air  in  the  same  direction,  and  it  only  becomes 
insensible  by  the  effect  of  a  stronger  wind.    Those  who  are  acquaint- 
ed with  the  vast  currents  which  prevail  in  seas,  will  easily  conceive 
that  they  must,  in  like  manner,  give  rise  to  considerable  motions  in 
the  atmosphere.     Miners  avail  themselves  of  this  principle  to  pro- 
duce a  current  of  air.^  In  an  adit,  through  which  a  stream  flows, 
they  place,  at  a  small  hilght  above  the  stream,  a  partition  of  boards. 
Below  this  the  air  follows  the  direction  of  the  stream,  above  it  lakes 
the  opposite  direction. 

355.  Many  mechanical  methods  have  been  devised  for  producing 
small  motions  in  the  air.     Of  this  kind  are  the  common  bellows, 
exhausting-pump,  condensing-pump,  &tc.     These    instruments  are 
made  of  different  forms  and  sizes  according  to  the  use  to  which  they 
are  applied.     Founders  employ  a  large  species  of  bellows,  and  they 
might  with  advantage  employ  a  condensing-pump.     Miners  some- 
times use  bellows  with  a  double  current  of  air,  and  exhausting- 
pumps. 

356.  As  air  is  put  in  motion  by  other  bodies,  so  also  it  may  itself 
put  other  bodies  in  motion,  both  solid  and  liquid. 

It  is  well  known,  that  in  hurricanes  the  air  has  power  to  tear  up 
trees,  destroy  houses,  and  raise  the  waves  of  the  sea  to  a  fearful 
height.  This  force  of  the  air  and  other  elastic  fluids  has  been  made 
use  of  for  various  purposes  in  the  arts.  The  pressure  of  a  moderate 
wind  puts  in  motion  the  sails  of  a  windmill.  The  steam  engine  is 
made  to  exert  an  enormous  power  by  jthe  force  of  aqueous  vapour  j 
and  is  employed  as  an  agent  in  giving  motion  to  all  kinds  of  ma- 
chinery. In  hydraulic  engines  which  act  by  impulse,  a  uniform  mo- 
tion is  obtained  by  the  condensation  of  air  in  a  reservoir.  In  the  air- 
gun,  it  is  the  condensation  of  this  fluid  which  produces  the  effect. 
In  fire  arms  the  force  is  derived  from  the  dilatation  of  gases  pro- 
duced by  the  inflammation  of  gunpowder.  It  is  in  vain  to  attempt  to 
enumerate  all  the  different  machines  which  depend  upon  the  moving 
force  of  elastic  fluids ;  and  posterity  will  still  find  ample  room  for 
important  inventions.  Among  the  machines  of  this  kind  which  are 
rather  amusing  than  useful,  may  be  mentioned  Hero's  fountain ; 
of  which,  however,  an  important  application  uas  been  made  in 
mines. 


Motions  of  Elastic  Fluids.  163 

357.  As  all  the  motions  produced  in  solids  and  liquids  take  place 
in  the  air,  the  theory  of  the  resistance  of  this  fluid,  is  a  very  import- 
ant subject ;  but  yet  very  difficult  to  investigate.     The  principles 
established  by  Newton  are  not  so  well  confirmed  by  experiment,  as 
his  laws  of  motion  in  the  case  of  solid  bodies. 

358.  It  is  the  uncertainty  of  this  theory  ^vhich  prevents  us  from 
fixing  completely  the  laws  of  the  descent  of  bodies  in  the  air.     What 
is  known  upon  this  subject  may  be  comprehended  in  the  following 
general  statement.     The  descent  of  bodies  in  the  air,  like  that  of 
bodies  in  a  liquid,  cannot  take  place  with  a  motion  uniformly  accele- 
rated ;  but  its  acceleration  must  decrease  in  the  same  manner  at 
each  instant.    Still  the  motion  cannot  become  uniform  as  in  a  liquid, 
because  the  density  of  the  air,  and  consequently  its  resistance,  con- 
tinually increase.     Thus,  if  we  suppose  a  body  to  fall  in  a  column 
of  air  of  sufficient  extent,  it  would  first  have  an  increasing  velocity, 
but  its  acceleration  would  continually  diminish.     At  a  certain  point 
the  acceleration  would  become  nothing,  and  the  velocity  would  be 
at  its  maximum.     Beyond  this  point,  the  resistance  always  increas- 
ing, the  velocity  itself  would  diminish,  till  at  length  this  also  would 
become  nothing,  and  the  body  would  remain  suspended  in  the  air. 
This  consequence  may  appear  paradoxical  to  one  who  has  not  an 
exact  idea  of  the  increase  of  the  density  of  the  air.     It  may  be 
shown  by  the  formula  for  measuring  heights  by  the  barometer,  that 
a  column  of  air  extending  into  the  interior  of  the  earth  to  the  depth 
of  700  miles,  would  be  100000  times  more  dense  than  that  at  the 
surface,  that  is,  five  or  six  times  more  dense  than  platina,  so  that  the 
heaviest  bodies  would  remain  suspended  in  it,  or  rather  would  have 
nn  upward  motion. 


SECTION  VL 

ELECTRICITY. 


CHAPTER  XXXII. 

Electrical  Machine,  and  General  Phenomena  of  Electricity. 

359.  THE  term  electricity  is  derived  from  the  Greek  word  elec- 
tron, signifying  amber,  on  account  of  the  property  winch  this  sub- 
stance was  known  to  possess  of  attracting  light  substances  when  rub- 
bed.    But  the  Greeks  did  not  suspect  that  this  phenomenon  was  the 
effect  of  a  very  remarkable  and  extensive  force  in  nature.     It  was 
not  till  the  17th  century  that  sulphur,  resins,  and  many  other  bodies, 
were  known  to  have  the  same  property  ;  the  inventor  of  the  air- 
pump  enriched  the  apparatus  of  the  philosopher,  with  a  second  very 
important  instrument,  called  the  electrical  machine.     The  limits  of 
this  work  will  not  permit  us  to  give  an  account  of  the  primitive  con- 
struction of  this  machine,  or  to  notice  the  various  improvements  it 
has  undergone.     We  can  only  mention  a  few  particulars  of  one  of 
the  most  improved  forms  of  these  machines. 

360.  The  two  most  essential  parts  are  the  plate  or  body  rubbed 
and  the  rubber.   The  body  rubbed  is  usually  a  circular  piece  of  pol- 
ished glass  ;  the  larger  it  is,  the  better.     This  plate  is  made  to  turn 
on  a  metallic  axis,  supported  by  a  wooden  frame.     Instead  of  a 
plate  a  glass  globe  or  cylinder  is  sometimes  used. 

The  rubber  for  the  most  part  consists  of  two  or  four  oblong  cush- 
ions, pressed  by  springs  against  the  glass,  so  as  to  produce  a  strong 
friction  when  the  plate  turns.  The  cushions  are  of  leather  filled 
with  hair,  and  placed  upon  metal  plates.  This  leather  is  to  be  cov- 
ered with  some  oily  substance,  over  which  should  be  spread  as 
equally  as  possible,  a  dry  amalgam  of  mercury  and  zinc.  To  each 


Electrical  Machine. 


165 


cushion,  and  on  the  side  towards  which  the  rotatory  motion  of  the 
plate  is  directed,  is  fitted  a  strip  of  gummed  taffeta,  or  oiled  silk, 
which  adheres  to  the  glass  when  the  machine  is  put  in  motion.  The 
frame  which  supports  the  cushions  should  be  of  metal ;  and  it  should 
be  attached  to  the  supports  of  the  plate,  not  by  metal,  but  by  firm 
glass  columns  or  tubes.  When  the  machine  is  in  operation  a  chain 
should  be  attached  to  the  metallic  part  of  the  cushions  at  one  ex- 
tremity, the  other  being  suffered  to  fall  upon  the  wooden  frame,  or 
what  is  still  better,  upon  the  ground.  This  circumstance  is  of  great 
importance,  for  the  effects  are  much  more  powerful  when  the  rubber 
has  a  metallic  communication  with  the  earth. 

361.  With  the  parts  of  the  machine  now  described,  we  can  ex- 
hibit to  the  senses  most  of  the  phenomena,  which  constitute  the  sci- 
ence of  electricity.  When  we  turn  the  plate,  the  atmosphere  being 
warm  and  dry,  we  observe  the  following  effects. 

(1.)  We  perceive  a  phosphoric  odour. 

(2.)  By  bringing  the  hand  or  face  gradually  near  the  plate,  we 
feel  at  a  certain  distance,  a  sensation  like  that  produced  by  the  con- 
tact of  a  spider's  web. 

(3.)  If  we  touch  the  plate  with  a  metallic  ball,  a  small  crackling 
spark  is  perceived,  and  if  the  hand  be  applied  instead  of  the  ball 
the  spark  is  accompanied  with  a  tingling  sensation. 

(4.)  In  the  dark,  this  phenomenon  is  much  more  striking  ;  and  as 
we  turn  the  machine,  streams  of  fire  are  seen  to  glance,  from  beneath 
the  gummed  taffeta  and  run  over  the  plate.* 


*  The  light  which  attends  the  electric  explosion,  appears  to  me  to 
be  the  simple  result  of  the  pressure  which  the  air  and  vapours  expe- 
rience, when  they  are  traversed  by  electricity.  For,  it  is  well  known 
that  simple  pressure  disengages  light  from  aeriform  fluids  ;  and,  on 
the  other  hand,  the  powerful  explosions  produced  by  electricity, 
prove  that  in  its  passage  through  bodies,  it  exerts  a  very  great  pres- 
sure. It  is  true  that  electric  light  is  observed  in  a  vacuum  ;  but 
what  we  here  call  a  vacuum,  is  simply  a  portion  of  space  occupied 
by  air  reduced  at  most  to  T|7  of  its  natural  density  ;  or  the  space 
may  be  filled  with  the  vapour  of  water  or  mercury.  If  denser  air 
emits  light  at  a  less  expression,  rarer  air  will  emit  it  when  an  infi- 
nitely greater  pressure  is  applied,  like  that  produced  by  the  rapid 
passage  of  electricity.  I  first  suggested  this  idea  in  an  account  of 
ray  experiments  upon  the  formation  of  water  by  simple  pressure. 


166  Electricity. 

(5.)  When  we  cease  to  turn  the  machine,  all  these  phenomena 
continue  for  some  time,  though  with  an  intensity  sensibly  decreasing  ; 
but  we  may  still  observe  phenomena  in  many  respects  the  most  im- 
portant, as  those  of  electric  attraction  and  repulsion.  The  plait;  in 
this  state  attracts  all  light  bodies,  retains  them  an  instant,  and  then 
repels  them.  If  we  bring  towards  the  plate,  balls  of  cork  suspended 
at  the  end  of  a  thread,  the  phenomenon  becomes  very  deserving  of 
attention.  If  the  thread  is  of  dry  silk,  the  small  ball  is  attracted, 
attaches  itself  for  a  moment  to  the  plate,  and  is  then  repelled.  This 
repulsion  is  durable ;  but  if  we  touch  the  small  ball,  it  is  again  at- 
tracted and  repelled.  On  the  contrary,  if  the  thread  is  of  linen,  and 
especially  if  it  be  moist ;  the  ball  will  only  be  attracted,  and  not  re- 
pelled. 

362.  When  a  body  manifests  these  phenomena  or  only  the  last 
which  we  have  described,  we  say  the  body  is  electrified  ;  and  the 
unknown  substance  which  produces  these  phenomena  is  called  the 
electric  matter  or  fluid. 

363.  It  still  remains  to  speak  of  an  important  part  of  the  electric 
machine,  the  conductor.     This  is  either  entirely  metallic,  or  at  least 
covered  with  a  metallic  substance,  as  gold  leaf,  for  example.     Its 
magnitude  and  form  are  arbitrary  ;  it  is  sometimes  a  ball,  commonly 
however  a  cylinder,  rounded  at  its  two  extremities.     For  a  plate  of 
two  feet  diameter,  we  make  it  about  three  feet  in  length  and  about 
six  inches  in  diameter.     It  communicates  with  the  plate  by  means  of 
two  rounded  branches  of  metal,  which  present  several  points  at  the 
distance  of  about  half  an  inch  from  the  plate,  at  the  part  where  the 
electricity  accumulates  and  passes  from  under  the  taffeta.     Care 
should  be  taken  that  there  be  no  other  points  or  prominent  angles. 
These  branches  should  be  disposed  in  such  a  manner  that  we  can 
remove  them,  and  substitute  one  for  the  other,  for  a  very  important 
purpose,  of  which  we  shall  speak  hereafter  ;  this  consists  in  making 
the  conductor  communicate  with  the  rubber  instead  of  the  plate. 
One  essential  circumstance  is,  that  the  conductor  be  placed  upon 
glass  supports,  and  have  no  other  communication  with  the  table  on 
which  it  rests. 

364.  When  we  turn  the  plate  the  conductor  becomes  electrified 
upon  its  whole  surface,  which  is  evinced  by  dfcthe  phenomena  men- 
tioned above  ;  only  the  spark  is  greater,  attended  with  a  louder  noise, 
and  is  more  sensibly  felt.  Accordingly  it  may  dart  to  the  distance 
of  several  inches,  especially  when  the  air  is  very  dry.  Another  very 


General  Phenomena  of  Electricity.  167 

remarkable  difference  is,  that  by  one  of  these  sparks  the  whole  elec- 
tricity of  the  conductor  is  taken  from  it  at  once,  whereas  the  plate 
3nly  loses  its  electricity  at  the  point  where  we  touch  it,  even  when 
the  motion  has  ceased.  If  the  conductor  communicate  with  the 
sarth  or  with  the  foot  of  the  machine,  by  a  brass  chain,  it  does  not 
uanifest  the  least  electricity  when  we  turn  the  plate. 

365.  These  experiments  clearly  show  the  different  properties  of 
;lass  and  metal  with  respect  to  electricity.     The  glass  is  electrified 
sy  friction.     It  strongly  retains  upon  its  surface  the  electricity  which 
s  accumulated  there,  and  suffers  it  to  be  taken  away  only  at  the 
precise  place  where  we  touch  it.     The  metal,  on  the  contrary,  is 
lot  electrified  by  friction  ;  it  receives  the  electricity  instantaneously 
;hrough  its  whole  extent,  when  it  is  placed  in  contact  with  the  glass  ; 
md  the  electricity  abandons  it  also  instantaneously  when  the  finger 
>r  some  metallic  body  is  presented  to  it. 

Hence  we  see  why  the  conductor  must  be  supported  by  .glass  col- 
amns,  and  why  it  is  not  electrified  when  it  communicates  with  the 
sarth.  The  metal  conducts  the  electric  matter,  and  the  glass  does 
lot. 

366.  Glass  and  metal  are  not  the  only  substances  which  exhibit 
these  opposite  properties  with  respect  to  electricity  ;  they  are  those 
in  which  the  properties  in  question  are  manifested  in  the  highest 
degree.* 

Under  this  point  of  view,  therefore,  we  may  divide  bodies  into  two 
ajreat  classes,  non-conductors,  and  conductors  of  electricity.  To  the 
first  class  belong ;  1 .  Among  inorganic  substances,  common  glass 
and  all  vitrifications  with  their  essential  constituent  principles  ;  earths 
and  metallic  oxydes,  and  all  natural  crystallizations  of  these  sub- 
stances: consequently,  all  precious  stones,  and  nearly  all  rough 
stones,  which  are  probably  only  collections  of  small  crystals.  Sul- 
phur and  atmospheric  air  belong  to  this  ckss.  Yet  the  latter  always 
has  some  conducting  power,  sometimes  weaker,  sometimes  stronger, 
according  to  the  quantity  of  water  it  contains.  2.  Most  of  the  dry 
animal  substances,  particularly  silk,  wool,  hair,  feathers.  3.  Many 
dry  vegetable  substances,  principally  the  resins  and  resinous  mix- 
tures, sealing-wax,  amber,  cotton,  paper,  sugar,  dry  wood,  especially 


*  The  resins  and  especially  gum  lac  bave  a  still  less  conducting 
power  than  glass ;  but  they  cannot  be  so  conveniently  employed  in 
machines  of  considerable  dimensions. 


168  Electricity. 

when  it  is  dried  by  the  fire ;  also  gross  oils.  Among  these  bodies, 
however,  there  is  no  one  that  does  not  conduct  electricity  to  a  cer- 
tain degree.  The  best  non-conductors  are  glass,  sulphur,  resin,, 
gum  lac,  silk  ;  the  rest  are  rather  to  be  considered  as  bad  conductors. 

The  best  conductors  are,  among  inorganic  bodies,  the  metals, 
water,  and  coal ;  among  organic  bodies,  living  animals  and  vegeta- 
bles ;  and  even  the  vegetable  fibre,  disengaged  from  all  the  oily  and 
resinous  parts,  appears  to  be  a  very  good  conductor,  at  least  this  is 
the  case  with  a  fibre  of  linen  ;  and  wood,  cotton,  &c.,  are  perhaps 
bad  conductors  only  on  account  of  the  oily  and  resinous  matter 
which  they  contain.* 

Non-conductors  are  also  called  electrics,  and  conductors  non-elec- 
trics ;  but  these  denominations  are  not  well  chosen. f 

367.  We  say  a  body  is  insulated  when  it  communicates  with 
other  visible  bodies  only  by  non-conductors.     To  insulate  a  body, 
therefore,,  we  suspend  or  support  it  by  one  of  these.    The  best  insu- 
lators are  glass,  sealing-wax,  silk,  and  wood  dried  by  the  fire.J 

368.  If  we  compare  the  phenomena  of  attraction  and  repulsion 
with  what  has  been  said  of  the  conducting  power  of  bodies,  we  de- 
duce the  following  important  principle  j 

Electrified  bodies  attract  those  which  are  not  electrified ;  bodies 
electrified  mth  the  same  electricity  repel  each  other. 

A  cork  ball  attached  to  a  non-conducting  silk  thread,  and  taken 
in  its  natural  state,  is  first  attracted  ;  but  is  repelled  as  soon  as  it  be- 
comes electrified,  and  this  repulsive  tendency  continues  until  the  ball 


*  It  is  very  desirable,  for  many  reasons,  that  skilful  chemists  would 
occupy  themselves  more  with  electricity.  What  has  been  said  of 
non-conductors  and  conductors,  affords  room  for  conjecture  that  there 
is  some  connexion  between  the  electric  properties  and  chemical 
composition  of  bodies,  which  mechanical  philosophy  alone  cannot 
discover. 

t  In  reality  the  distinction  of  non-conductors  and  conductors  is 
not  much  better.  All  bodies,  even  gum  lac,  can  be  penetrated  by 
powerful  electricity  ;  so  that  all  these  distinctions  are  to  be  regarded 
as  merely  relative,  and  none  as  absolute. 

|  Nothing  insulates  better  than  a  cylinder  of  gum  lac.  Coulomb 
proxed  that  a  thread  of  gum  lac,  drawn  out  by  the  flame  of  a  candle, 
is  nearly  a  perfect  insulator  when  the  quantity  of  electricity  is 
small. 


General  Phenomena  of  Electricity.  169 

has  lost  its  electricity  by  contact  with  an  uninsulated  conductor.  If, 
on  the  contrary,  it  is  suspended  by  a  conducting  linen  thread,  and 
not  insulated,  it  cannot  be  saturated  with  electricity,  and  for  this  rea- 
son it  is  constantly  attracted. 

369.  Upon  these  phenomena  of  electric  attraction  and  repulsion 
are  founded  almost  all  the  instruments  which  pass  under  the  name 
of  electroscopes  and  electrometers.  These  serve  to  measure  the 
intensity  of  electricity,  but  for  the  most  part  they  answer  their 
purpose  imperfectly.  Still  they  are  useful  in  many  experiments, 
and  therefore  we  shall  give  a  brief  description  of  them.  The  most 
simple  is  the  thread  electrometer.  Two  small  balls  of  cork  or  elder 
pith  are  attached  to  the  extremities  of  a  linen  thread.  We  suspend 
them  to  the  conductor  or  some  other  electrified  body,  so  that  in  their 
natural  position  they  touch  each  other  by  the  effect  of  gravity.  As 
soon  as  they  are  electrified  they  diverge,  and  their  divergence  is  pro- 
portional to  the  intensity.  The  size  of  this  instrument  varies  accord- 
ing to  the  use  to  which  it  is  to  be  applied.  For  minute  degrees  of 
electricity  they  must  be  v>  ry  small.  Most  of  the  electrometers  are 
only  modifications  of  this .  We  have  not  room  for  particular  descrip- 
tions ;  and  can  only  observe  that  the  best  are  the  jar  electrometer  of 
Cavallo,  the  air  electrometer  of  Saussure,  the  gold-leaf  electrometer 
of  Bennet,  and  the  straw  electrometer  of  Volta.*  We  cannot,  how- 
ever, omit  the  quadrant  electrometer  of  Henley,  because  it  is  con- 
sidered as  an  essential  appendage  to  an  electrical  machine.  A 
graduated  semicircle  is  attached,  by  its  diameter,  to  a  column  of 
metal  or  undried  wood,  so  as  to  be  at  a  small  distance  from  the 
column,  the  diameter  being  parallel  to  it.  To  the  centre  is  attached 
a  small  and  very  moveable  pendulum,  made  of  a  stem  of  whale- 
bone, and  having  a  small  cork  ball  at  its  extremity.  The  column, 
which  is  much  longer  than  the  pendulum,  may  be  screwed  by  its 
lower  extremity,  perpendicularly  to  the  conductor  of  the  machine ; 
or  it  may  be  supported  upright  by  a  metallic  rod,  and  be  removed 
at  pleasure.  This  electrometer  being  placed  on  the  conductor  re- 
ceives its  electricity  ;  and  as  the  pendulum  and  column  are  electri- 
fied in  the  same  way,  the  pendulum  is  repelled  by  the  column,  and 


*  It  is  surprising  that  the  author  has  not  mentioned  the  electric 
balance  of  Coulomb  ;  the  only  one  which  gives  the  exact  measure  of 
electricity.  We  shall  speak  of  it  hereafter. 

Elem.  22 


170  Electricity. 

the  height  to  which  it  rises  on  the  graduated  quadrant,  indicates  the 
intensity  of  the  electricity. 

370.  The  conducting  power  of  bodies  does  not  depend  upon  their 
material  constitution  alone,  but  also  on  their  form.     If,  while  the 
plate  is  in  motion,  we  bring  a  pointed  body  of  any  substance  what- 
ever, near  the  conductor,  its  conducting  power  becomes  obvious  at 
a  considerable  distance.     Metallic  points  manifest  this  effect  in  the 
highest  degree.     This  conducting  power  of  points  is  also  manifest, 
when  we  attach  a  point  to  the  conductor  in  such  a  manner  that  the 
sharp  part  is  directed  from  the  conductor  into  the  air.     It  is  then 
impossible  to  charge  the  conductor  with  a  high  degree  of  electricity. 

On  the  contrary,  the  dispersion  of  electricity  becomes  the  more 
difficult,  in  proportion  as  the  body  is  larger  and  rounder.  In  this 
case  it  is  necessary  to  bring  the  bodies  much  nearer,  and  the  pas- 
sage is  accompanied  by  a  spark. 

371.  When  this  effluent  current  of  electricity  leaves  the  points,  a 
certain  motion  is  always  produced  in  the  air,  which  may  be  rendered 
visible  by  means  of  the  flame  of  a  candle,  or  some  vapour.     Upon 
this  motion  is  founded  the  electric  mill,  which  consists  of  a  strip  of 
copper  in  the  form  of  an  S,  aud  carefully  sharpened  at  both  ends, 
which  is  made  to  turn  circularly  upon  a  point  placed  at  its  centre. 
When  this  point  is  screwed  to  the  conductor,  and  the  machine  is  put 
in  motion,  the  wheel  turns  backward  with  great  velocity.     It  is  im- 
portant to  remark,  that  the  motion  of  the  air  is  always  directed  to 
the  sharp  part  of  the  point. 

372.  As  we  can  cause  the  electricity  of  the  plate  to  pass  to  the 
principal  conductor,  so  also  this  may  be  transmitted  to  a  second  con- 
ductor provided  it  be  insulated.     Thus,  for  example,  a  man  may  be 
electri6ed  when  he  stands  upon  an  insulator,  which  is  usually  a  stool 
supported  by  four  glass  feet.     In  this  condition  his  body  exhibits 
all  the  electric  phenomena,  without  his  experiencing  any  particular 
sensation,  except  when  sparks  are  taken  from  his  body. 

373.  The  chemical  effects  of  electricity  are  extremely  remark- 
able.    We  shall  only  mention  the  inflammation  of  alcohol  and  deto- 
nating gas*  by  the  electric  spark.     In  the  sequel  we  shall  be  made 
acquainted  with  other  phenomena  of  the  same  kind.     We  shall 
speak,  in  another  place,  of  the  electrical  appearances  exhibited  in 
the  dark  and  in  rarefied  air. 

*  That  is  a  mixture  of  two  parts  by  bulk  of  hydrogen  and  one  of 
oxygen. 


Opposite  Electricities.  171 

CHAPTER  XXXIII. 

Opposite  Electricities. 

374.  Iif  the   first  half  of  the   last  century,   Dufay,   a   French 
philosopher,   discovered  that  there  were  two  kinds  of  electricity, 
which,  when  considered  separately,  have  the  greatest  resemblance, 
but   when   compared   together  exhibit  opposite   phenomena.     He 
named  the  one  vitreous  and  the  other  resinous  electricity,  because 
the  first  was  obtained  from  glass,  and  the  second  from  resin.*  After 
his  death,  philosophers  seemed  to  forget  this  nice  distinction,  the 
discovery  of  which  does  so  much  honour  to  the  sagacity  of  Dufay. 
At  length,  in  the  latter  part  of  the  last  century  the  celebrated  Frank- 
lin continued  the  investigation,  and  pointed  out  so  perfectly  the  dif- 
ference between  the  two  electricities,  that  this  discovery  has  since 
become  the  key  to  the  most  remarkable  electrical  phenomena.     In- 
stead of  the  names  vitreous  and  resinous,  chosen  by  Dufay,  Frank- 
lin adopted  those  of  positive  and  negative  ;  whence  it  has  become 
common  to  indicate  the  one  by  the  sign  -f-  E,  and  the  other  by  the 
sign  —  JE.    Yet  as  the  denomination  of  Dufay  is  founded  upon  fact, 
and  that  of  Franklin  upon  mere  hypothesis,  and  one  too  which  has 
lost  much  of  its  plausibility  since  his  time  ;  the  terms  of  Dufay  are 
to  be  preferred. 

375.  It  is  now  known  that  the  two  kinds  of  electricity  may  be  ex- 
cited in  many  ways,  and  that  they  are,  in  fact,  both  produced  at  the 
same  time,  one  in  the  rubber,  and  the  other  in  the  body  rubbed. 
Thus  when  the  conductor  is  disposed  in  the  manner  described  in 
the  preceding  chapter,  it  is  as  easy  to  charge  it  with  the  resinous  as 
with  the  vitreous  electricity.     It  is  only  necessary  to  insulate  the 

*  The  definition  is  not  strictly  exact.  Glass  rubbed  with  wool, 
takes  the  electricity  called  vitreous  ;  and  when  rubbed  with  cat 
skin,  the  resinous.  I  know  of  no  body  which  may  not  be  made  to 
take  both  electricities,  by  changing  the  rubber,  or  by  some  slight 
modification  in  the  circumstances  of  the  body  rubbed.  Still  the  dis- 
tinction between  the  two  electricities  is  not  the  less  real,  because  it 
rests  upon  the  attractions  and  repulsions  which  belong  to  them,  and 
not  upon  the  nature  of  the  bodies  which  produce  them. 


172  Ekctricity. 

rubber,  and  make  the  conductor  communicate  with  it ;  then  to 
the  plate  in  communication  with  the  ground,  or  draw  from  it  contir 
ually,  by  means  of  points  properly  placed,  the  vitreous  electricit) 
produced  on  its  surface. 

376.  When  we  turn  the  plate,  after  having  made  this  change,  tl 
conductor  which  is  then  in  communication  with  the  body  rubbed, 
and  which  itself  makes  a  part  of  it,  becomes  electrified  and  exhibits 
all  the  phenomena  mentioned  in  the  preceding  chapter.     Only  the 
electricity  is  always  much  more  feeble,  which  is  probably  nothing 
more  than  an  accidental  circumstance,  arising  from  this,  that  the  vit- 
reous electricity  which  passes  from  the  conductor  to  the  plate,  and 
which  counteracts  the  effects  of  the  other,  cannot  be  entirely  taken 
from  the  plate,  as  fast  as  it  becomes  fixed  there. 

377.  The  principal  difference  between  the  two  electricities,  is 
seen  in  the  phenomena  of  attraction  and  repulsion  ;   for  two  bodies 
which  repel  each  other  when  they  have  the  same  electricity,  attract 
each  other  when  they  have  opposite  electricities ;  from  which  we 
deduce  the  following  law  ; 

Electricities  of  the  same  name  repel  each  other  ;  and  electricities  oj 
opposite  names  attract  each  other. 

To  be  convinced  of  the  exactness  of  this  law,  we  dispose  the  ap- 
paratus in  the  manner  above  described,  so  as  to  give  resinous  elec- 
tricity to  the  conductor,  the  plate  always  absorbing  the  vitreous. 
The  latter  being  a  non-conducting  body,  always  retains  a  small  por- 
tion of  this  electricity,  in  spite  of  our  efforts  to  remove  it.  Then 
we  take  a  cork  ball  suspended  by  a  silk  thread  ;  if  we  bring  this 
near  the  conductor,  it  is  attracted  by  it,  and  being  saturated  with 
resinous  electricity  is  again  repelled  ;  but  in  this  state  it  is  attracted 
by  the  glass  plate  ;  its  resinous  electricity  is  destroyed  ;  after  which 
it  charges  itself  with  vitreous  electricity,  and  is  then  repelled  by  the 
plate.  In  this  state,  it  is  again  attracted  by  the  conductor  ;  and  thus 
a  position  may  be  easily  found,  in  which  the  ball  is  alternately  at- 
tracted and  repelled  by  each,  so  as  to  keep  up  a  constant  vibration. 

378.  It  is  evident  from  this  experiment,  that  one  electricity  des- 
troys the  other.     This  becomes  still  more  manifest  when  we  dispose 
the  machine  so  as  to  produce  the  vitreous  electricity,  with  the  sin- 
gle precaution  of  insulating  the  rubber,  and  causing  it  to  communi- 
cate by  a  chain  with  the  conductor.     In  this  case  we  do  not  find  the 
least  trace  of  electricity  in  the  conductor.    Generally  when  we  unite 
unequal  degrees  of  the  two  electricities,  the  least  intense  is  always 


Opposite  Electricities.  173 


destroyed,  and  the  most  intense  diminished.  Heiice  we  see  why  it 
s  always  necessary  that  the  plate  communicate  with  the  conductor 
>vhen  the  machine  is  fitted  up  for  vitreous  electricity ;  an^  why  it 
nust  communicate  with  the  rubber  when  the  machine  is  intended  to 
iroduce  resinous  electricity.* 

379.  From  these  relations  between  the  two  electricities,  we  de- 
luce  a  mode  of  distinguishing  them.  The  apparatus  commonly 
ised  for  this  purpose,  consists  of  a  thread  electrometer,  suspended 
md  insulated,  and  a  stick  of  sealing-wax.  We  know  from  experi- 
nent  that  wax  rubbed  with  wool,  leather,  and  linen,  always  acquires 
he  resinous  electricity.  We  communicate  to  the  electrometer  the 
electricity  which  we  wish  to  examine,  so  that  the  balls  repel  each 
>ther,  and  continue  some  time  separated  ;  then  we  bring  the  rubbed 
lealing-wax  near  them  ;  if  the  electrometer  has  resinous  electricity, 
L  part  of  this  electricity  is  disguised  by  that  of  the  wax,  and  the 
breads  collapse  ;  if,  on  the  contrary,  it  has  vitreous  electricity,  they 
diverge  more  than  before. 


Electrical  Phenomena  in  the  Dark  and  in  Rarefied  Air. 

380.  In  the  dark  the  two  electricities  are  distinguished  from  each 
)ther  in  a  remarkable  manner  ;  that  is,  by  a  difference  in  the  lumi- 
IGUS  appearances  which  are  presented  when  the  electricity  is  taken 
iway  by  points.  If  the  conductor  is  charged  with  vitreous  electri- 
city, and  a  point  is  brought  near  it,  we  see,  at  a  considerable  dis- 


*  Generally  when  two  insulated  bodies  are  electrified  by  their 
nutual  friction,  one  takes  the  vitreous  and  the  other  the  resinous 
;lectricity.  This  happens,  therefore,  to  the  plate  and  rubber,  when 
hey  are  insulated.  It  takes  place  also,  when  the  rubber,  always  in- 
sulated, communicates  with  the  conductor.  But  the  effect  soon  has 
i  limit ;  for  the  vitreous  electricity  which  accumulates  on  the  plate, 
lot  being  able  to  escape,  prevents  new  quantities  of  vitreous  electri- 
city from  flowing  there,  and  consequently  from  becoming  apparent. 
Whereas  if  we  take  away  the  vitreous  electricity  by  points,  as  fast 
\s  it  is  produced,  then  the  natural  electricity  of  the  conductor  is  de- 
composed, and  the  vitreous  electricity  is  accumulated  on  the  plate 
ivithout  interruption  ;  and  reciprocally  the  conductor  is  in  a  durable 
ind  inc.ieasing  state  of  vitreous  electricity. 


174  Electricity. 

tance  a  bright  star,  at  the  extremity  of  the  metallic  point  which  be- 
comes more  brilliant  as  the  point  is  brought  nearer.  If  the  point 
be  fastoied  to  the  conductor,  and  the  hand  or  some  other  conduct- 
ing body  be  brought  near,  we  no  longer  see  a  bright  star,  but  a 
pencil  of  diverging  rays.  On  the  contrary,  if  the  conductor  is  charg- 
ed with  resinous  electricity,  the  two  phenomena  occur  in  the  inverse 
order. 

381.  We  shall  add  another  experiment,  which  does  not,  indeed, 
show  so  clearly  the  difference  between  the  two  electricities,  but 
which  is  remarkable  in  other  respects. 

Although  dry  atmospheric  air  is  a  bad  conductor  of  electricity,  yet 
highly  rarefied  air  is  capable  of  being  traversed  by  it ;  this  is  evinced 
by  the  following  experiment.  •  We  rarefy  the  air  in  a  glass  vessel 
having  a  metallic  cover,  as  the  receiver  of  an  air-pump,  or  a  glass 
tube  prepared  for  the  purpose  ;  then,  if  we  make  the  conductor 
communicate  with  one  extremity  of  the  vessel,  and  the  other  with  the 
ground,  it  being  dark,  the  passage  of  the  electricity  through  the  rare- 
fied air  takes  place  under  the  form  of  a  whitish  light.  This  phe- 
nomenon, which  may  be  called  the  electrical  aurora  borcalis,  con- 
tinues as  long  as  we  turn  the  plate. 

There  are  many  ways  of  varying  this  experiment,  and  most  of 
them  exhibit  very  agreeable  luminous  appearances.  If  we  place  at 
the  two  extremities  of  the  vessel  two  metallic  points  directed  inward, 
the  light  escapes  from  one  diverging,  and  enters  the  other  converg- 
ing. If  the  vessel  be  a  receiver,  and  a  metallic  rod  be  transmitted 
through  the  upper  cover,  having  at  its  end  metallic  radii  in  the  form 
of  a  star,  placed  horizontally,  the  electric  light  passes  off  from  each 
of  the  points  toward  the  lower  plate,  producing  the  appearance  of  a 
fountain  of  fire.  By  substituting  for  the  star  a  ring  or  a  body  of  any 
other  figure,  different  forms  may  be  given  to  the  current  of  light. 

382.  In  these  experiments  the  following  circumstance  is  worthy 
of  attention.    When  we  bring  the  conductor  near  the  vessel  in  which 
the  electrical  light  is  exhibited,  a  particular  movement  takes  place  in 
this  light  at  the  place  to  which  the  conductor  is  brought.     We  can 
then  take  sparks  from  the  conductor,  but  they  are  variable  as  to  their 
intensity.     This  observation  proves  that  electric  attraction  acts  even 
through  the  glass,  though  a  non-conductor. 

We  remark,  in  this  connexion,  that  the  phosphorescence,  which 
is  exhibited  by  some  barometers,  not  perfectly  deprived  of  air,  when 
we  incline  the  tube  in  the  dark  so  as  to  make  the  mercury  pass  from 


Hypothesis  of  Franklin.  175 

one  end  to  the  other,  has  its  cause  in  this  same  electrical  phenome- 
non. 

383.  These  luminous  appearances  are  the  same,  or  nearly  the  same, 
with  respect  to  both  electricities.   Franklin  and  many  of  his  followers 
believed  that  there  was  a  difference,  consisting  in  this,  that  when  the 
conductor  is  charged  with  vitreous  electricity,  the  electricity  passes 
always  from  the  conductor ;  and  that  when  it  is  charged  with  resinous 
electricity,  the  electric  light  passes  from  the  ground  to  the  conductor. 
But  this  was  rather  a  deduction  from  the  hypothesis  than  from  actual 
observation  ;  for,  with  the  most  exact  attention  to  all  the  motions  of 
the  electric  light,  it  is  impossible  to  determine  the  directions  of  this 
motion,  because  it  takes  place  so  instantaneously.     It  appears  to  an 
attentive  eye,  sometimes  to  approach  the  conductor  and  sometimes 
to  recede  from  it,  and  sometimes  to  be  directed  both  ways  at  the 
same  time. 

Hypothesis  of  Franklin. 

384.  According  to  Franklin,  electric  phenomena  are  the  effect  of 
a  single  substance  infinitely  subtile,  which  is  diffused  through  all 
bodies,  by  laws  analogous  to  those  of  caloric.     Its  particles  repel 
each  other,  but  they  are  more  or  less  attracted  by  other  bodies.    As 
long  as  this  electric  matter  is  in  a  state  of  equilibrium  in  a  system  of 
bodies,  no  electric  phenomenon  is  exhibited  ;  but  when,  on  the  con- 
trary, this  matter  is  accumulated  above  or  diminished  below  the 
point  of  equilibrium,  the  body  is  electrified,  positively  in  the  first 
case,  negatively  in  the  second  ;  and  the  electric  phenomena  are  pro- 
duced by  the  efforts  which  the  electric  matter  makes,  to  restore  the 
equilibrium  which  has  been  disturbed. 

385.  What  principally  led  Franklin  to  this  hypothesis,  was  the 
observation  that  the  rubber  must  not  be  insulated ;  for,  in  fact,  we 
are  obliged  to  admit  that  the  electricity  which  accumulates  on  the 
plate,  passes  from  the  rubber.     The  possibility  of  charging  the  con- 
ductor successively  with  the  two  electricities,  is  easily  explained 
upon  this  hypothesis,  which  indeed  affords  a  solution  to  a  great  pro- 
portion of  electric  phenomena.     But  the  subject  of  attraction  and 
repulsion  presents  a  difficulty.     If  two  bodies,  electrified  positively, 
are  brought  together,  according  to  this  hypothesis,  their  surrounding 
electric  atmospheres,  being  pressed,  force  them  to  recede.     If  two 
bodies  negatively  electrified,  are  brought  near  each  other,  the  natural 


176  "Electricity. 

I 

electricity,  placed  between  their  rarefied  atmospheres,  and  thereby 
rendered  more  intense,  is  forced  to  expand,  and  thus  obliges  tho 
bodies  to  diverge.  Lastly,  if  two  bodies,  one  positively,  the  other 
negatively  electrified,  are  brought  towards  each  other,  they  approach, 
because  the  positive  atmosphere  of  the  first,  is  attracted  by  the  nega- 
tive atmosphere  of  the  second.  We  shall  find  several  phenomena 
hereafter  which  admit  of  only  a  forced  explanation  upon  this  hypoth- 
esis ;  among  others  the  following  may  be  mentioned ;  in  the  pas- 
sage of  electricity  through  a  point,  the  current  of  air  which  it  pro- 
duces is  always  directed  towards  the  sharp  part  of  the  point ;  and 
it  is  still  the  same  when  we  electrify  the  conductor  negatively, 
according  to  the  expression  of  Franklin.  For  a  circumstantial  ac- 
count of  this  hypothesis,  we  refer  the  reader  to  Franklin's  letter  on 
electricity. 


Hypothesis  of  Synnner. 

3S6.  Robert  Symmer,  in  the  first  part  of  the  51st  volume  of  the 
Philosophical  Transactions,  published  another  hypothesis,  which  has 
taken  the  place  of  that  of  Franklin,  in  the  minds  of  most  inquirers, 
because  it  satisfies  the  phenomena  better.  According  to  Symmer 
there  are  two  kinds  of  electric  matter  which  attract  each  other,  while, 
on  the  contrary,  the  particles  of  each  taken  separately,  repel  each 
other.  Their  union,  which  is  called  combined  electricity,  produces 
the  state  of  equilibrium  ;  their  disunion  the  electric  state.  The  one 
taken  separately  gives  the  phenomena  of  vitreous  electricity,  the 
other  those  of  resinous  electricity.  When  two  bodies  have  both  vit- 
reous, or  both  resinous  electricity,  their  homogeneous  atmospheres 
repel  each  other.  When  one  has  vitreous  and  the  other  resinous, 
their  heterogeneous  atmospheres  attract  each  other. 

The  combined  electricity  of  the  rubber  is  decomposed  by  friction  ; 
the  plate  attracts  the  vitreous  electricity  ;  the  resinous  electricity,  be- 
coming free,  escapes  into  the  earth  through  the  conductor  attached 
to  the  rubber ;  new  currents  of  combined  electricity  flow  through 
this  same  conductor ;  and  as  long  as  the  machine  is  in  motion,  this 
effect  continues  in  an  uninterrupted  manner.  This  hypothesis  ex- 
plains also  without  difficulty,  some  more  complicated  phenomena, 
which  will  present  themselves  in  the  sequel. 


Electric  Balance.  177 

387.  The  earliest  hypotheses  are  entirely  inadmissible.  Among 
the  more  recent,  that  of  Deluc  which  may  be  found  in  his  New 
Ideas  upon  Meteorology,  deserves  attention. 

it  may  be  doubted  whether  any  of  these  hypotheses  is  entirely 
conformable  to  fact ;  but  that  of  Symmer  is  unquestionably  entitled 
to  the  preference,  because  it  affords  the  best  explanation  of  electrical 
phenomena. 


Addition. 
The  Electric  Balance. 

388.  M.  Coulomb  has  greatly  improved  upon  the  ideas  of  Symmer. 
He  is  the  first  who  may  be  said  to  have  reduced  them  to  an  exact 
theory,  especially  by  the  discovery  which  he  made  of  the  law  of 
electric  attractions  and  repulsions  ;  for  it  is  not  enough  that  the  strict 
inquirer  knows  bodies  to  attract  or  repel  each  other  in  a  given  man- 
ner.    If  he  would  adapt  an  hypothesis  to  these  facts,  this  hypothesis 
must  be  such  as  will  represent  them  with  exactness ;  that  is,  such 
that  all  the  circumstances  of  the  phenomena  may  be  deduced  from 
it  by  rigorous  calculation.     This  is  effected  by  the  discovery  of  M. 
Coulomb.     In  order  to  render  it  intelligible,  we  must  describe  the 
instrument  with  which  it  was  made,  and  which  M.  Coulomb  called 
the  electric  balance. 

389.  To  a  very  fine  silver  wire  fixed,  at  one  of  its  extremities,  to 
some  solid  body,  we  suspend  a  long  and  slender  needle  of  gum  lac,  a 
substance  which  strongly  resists  the  passage  of  electricity.    We  place 
this  needle  in  a  horizontal  position,  and  adapt  to  one  end  a  very  small 
circle  of  gilt  paper.     This  circle  is  the  body  to  which  we  communi- 
cate electricity  ;  the  needle  of  gum  lac  serves  to  insulate  it ;  and  the 
silver  wire,  by  its  force  of  torsion,  serves  to  measure  the  attractive  or 
repulsive  force  which  is  exerted  upon  it  by  the  electrified  bodies  pre- 
sented to  it. 

It  is  obvious  that  only  a  very  small  force  is  necessary  if  it  be  de- 
terminate and  constant,  to  twist  this  wire  through  360°  or  180°,  and 
consequently  to  derange,  to  the  same  degree,  the  natural  state  of  equi- 
librium of  the  needle.  Accordingly,  the  angle  described  by  the 
paper  circle  will  increase  in  proportion  to  the  intensity  of  the  attrac- 
tion or  repulsion  ;  and  if  we  know  the  force  of  torsion  corresponding 

Elem.  23 


179  Electricity. 

• 

to  different  angles  of  deviation,  we  may  easily  determine  how  the 
electric  action  varies  with  the  distance.  Now  this  may  be  easily 
done  5  for  M.  Coulomb  has  proved,  by  very  exact  experiments,  that 
the  force  of  torsion  in  a  wire  of  a  certain  length,  is  exactly  propor- 
tional to  the  angle  of  torsion ;  and  to  avoid  the  difference  which 
might  arise  from  the  irregular  form  of  bodies,  we  employ  for  the  at- 
tracting or  repelling  body,  a  copper  sphere  insulated  at  the  extremity 
of  a  cylinder  of  gum  lac.  The  small  gilt  circle  upon  which  it  is  to 
act,  may  be  considered  as  a  point ;  the  sphere  is  so  placed  that  in 
lhe  natural  position  of  equilibrium,  it  is  in  contact  with  the  circle  ; 
and  to  avoid  the  errors  produced  by  agitations  in  the  air,  the  whole 
apparatus  is  inclosed  in  a  glass  case,  on  the  outside  of  which  are 
traced  the  horizontal  divisions,  which  serve  to  measure  the  angles 
described  from  the  point  of  contact. 

Let  us  now  consider  the  mode  of  using  this  instrument. 
390.  We  take  the  copper  sphere  by  its  handle  of  gum  lac,  and 
communicate  to  it  a  certain  quantity  of  electricity  from  the  conductor, 
after  which  we  replace  it  in  the  balance.  The  gilt  paper  shares  its 
electricity  and  is  immediately  repelled.  After  a  series  of  oscilla- 
tions the  needle  settles  down  at  a  determinate  angular  distance  from 
the  sphere,  in  which  position  it  is  evident  that  the  torsion,  experienced 
by  the  wire,  is  in  equilibrium  with  the  repulsive  force,  and  may 
thus  serve  to  measure  it.  In  order  to  be  definite,  let  us  suppose 
that  this  torsion,  or  the  repulsive  force  which  acts  upon  the  needle, 
is  36o. 

Then  if  we  forcibly  twist  the  wire  in  a  contrary  direction,  which 
may  be  done  by  means  of  an  index  placed  at  the  top  of  the  glass 
cover,  and  attached  to  the  wire,  the  torsion  will  preponderate  and 
the  circle  will  approach  the  sphere.  Suppose  that  we  turn  the  in- 
dex till  the  needle  is  only  18°  from  the  sphere,  instead  of  36°, 
as  it  was  at  first ;  we  find  that  in  order  to  bring  it  into  this  position 
it  is  necessary  to  turn  the  index  126°.  This  torsion  is  evidently  to 
be  added  to  the  preceding  36°,  and  in  the  second  case,  the  total 
force  of  torsion  would  be  162°,  if  the  small  circle  were  in  the  same 
position  as  before  ;  but  as  the  angular  distance  of  the  needle  is  18° 
less,  the  wire  is  evidently  untwisted  by  this  quantity  ;  the  actual  tor- 
sion is  therefore  162°  —  18°  or  144o. 

By  comparing  these  results  we  find  that  when  the  deviations  of 
ie  needle  were  30°  and  18°,  the  forces  of  torsion  required  to  coun- 
terbalance the  repulsive  force,  or  in  other  words,  the  intensities  ot 


Striking  Distance.  179 

this  repulsive  force,  are  represented  by  36°  and  144°  ;  whence  it 
follows  that  if  the  deviations  of  the  needle  are  as  2  to  1,  or  as  1  to  |, 
the  repulsive  forces  are  as  1  to  4  j  that  is,  the  repulsive  force  of 
the  electricity  increases  as  the  square  of  the  distance  diminishes,  or 
diminishes  as  the  square  of  the  distance  increases.  In  general,  it  is 
inversely  proportional  to  the  square  of  the  distance.  Other  experi- 
ments, made  in  the  same  manner,  with  different  intensities,  give  the 
same  ratio.  If  we  apply  the  method  to  electric  attractions,  we  find 
them  subject  to  the  same  law. 

391.  For  the  sake  of  greater  simplicity  we  here  suppose  the  dis- 
tance between  the  sphere  and  paper  circle  to  be  measured  by  the  arc 
of  the  circle  which  separates  them  ;  this  is  not  rigorously  exact,  for  it 
is  the  chord  which  measures  the  distance.     But  when  the  arcs  are 
small,  as  we  suppose  them  in  this  example,  the  difference  is  incon- 
siderable ;  besides,  allowance  is  made  for  it  in  nice  calculations ; 
and  it  is  only  when  this  correction  is  made  that  the  preceding  law  is 
exact.     It  is  remarkable  that  it  is  the  same  as  that  of  the  celestial 
attractions. 

392.  By  means  of  what  precedes,  the  theory  of  the  two  fluids,  as 
M.  Coulomb  presents  it,  may  be  reduced  to  this  hypothesis. 

Electrical  phenomena  are  produced  by  the  reciprocal  action  of 
tivo  invisible  and  imponderable  fluids,  the  properties  of  which  are, 
that  the  particles  of  each  repel  one  another,  and  attract  those  of  the 
opposite  fluid,  in  the  inverse  ratio  of  the  square  of  the  distance. 

By  this  hypothesis  we  can  represent  all  the  phenomena,  and  sub- 
ject many  of  them  to  a  rigorous  calculation  ;  but  we  must  not  regard 
it  as  any  thing  more  than  a  convenient  mode  of  explanation ;  we 
only  know  that  the  phenomena  take  place  as  if  they  were  produced 
by  two  fluids  endowed  with  the  above  properties  j  the  actual  nature 
of  electricity  is  still  unknown. 


CHAPTER  XXXIV. 

Striking  Distance,  Sphere  of  Activity,  Accumulated  Electricity. 

398.  WHEN  we  bring  a  conducting  body,  not  pointed,  near  the  elec- 
trified conductor,  we  have  seen  that  at  a  certain  distance  a  passage 
of  electricity  takes  place  by  means  of  a  spark,  and  the  same  sort  01 


180  Electricity. 

electricity  which  the  conductor  possesses,  is  thus  communicated  to 
the  body ;  and  the  body  is  said  to  be  electrified  by  communication. 
If  the  body  which  receives  the  electricity  is  well  insulated,  it  retains 
its  electricity  when  the  conductor  is  removed.  The  space  surround- 
ing the  electrified  body,  within  which  this  effect  is  produced,  is  called 
its  striking  distance  ;  and  we  say  that  a  conductor  gives  sparks  at  4 
or  6  inches,  when  the  passage  is  effected  at  these  distances.  The 
Striking  distance  is  very  variable.  It  depends  on  the  intensity  of  the 
electricity,  the  conducting  power  of  the  body,  the  form  of  this  body, 
and  the  qualities  of  the  surrounding  air. 


Sphere  of  Activity. 

394.  The  action  of  electricity  is  not  limited  to  its  striking  distance  ; 
it  manifests  itself  beyond  this  distance  in  a  manner  less  remarkable, 
it  is  true,  but  still  perhaps  not  less  worthy  of  attention.  The  law  of 
this  action  is  very  clearly  demonstrated  in  the  following  manner. 

We  take  an  insulated  conductor,  a  metallic  tube,  for  example, 
and  apply  to  one  of  its  extremities  a  sensible  electrometer,  present- 
ing the  other  extremity  to  the  conductor  of  the  machine.  We  then 
perceive,  by  the  electrometer,  that  the  insulated  conductor  gives 
signs  of  electricity  beyond  the  striking  distance,  and  always  of  the 
same  electricity  with  that  of  the  conductor  of  the  machine.  But  this 
electricity  is  distinguished  from  that  communicated  by  the  spark  ; 
thus,  if  we  remove  the  body  from  the  conductor,  it  diminishes  in  the 
same  manner  as  it  before  increased ;  whereas  the  other  suffers  no 
diminution  by  a  change  of  place,  except  what  is  inevitably  produced 
by  the  conducting  power  of  the  air.  This  mode  of  electrifying  a 
body  is  called  developement  of  electricity.  If  we  again  bring  the 
conductor  to  the  point  where  its  shows  sensible  signs  of  electricity, 
and  then  touch  it  with  the  finger,  the  threads  of  the  electrometer 
collapse,  and  all  traces  of  electricity  disappear.  But  if  we  remove 
the  conductor  a  little,  always  keeping  it  insulated,  the  threads  again 
diverge,  and  always  in  proportion  to  the  distance  to  which  it  is  re- 
moved. This  phenomenon,  precisely  opposite  to  what  happened 
before  we  touched  the  conducting  body,  indicates  that  the  body  has 
passed  to  the  opposite  state  of  electricity,  which  indication  is  con- 
firmed by  the  trial  electrometer. 


Sphere  of  Activity.  181 

395.  The  whole  space  within  which  this  effect  is  produced,  is 
called  the  sphere  of  activity  ;  and  the  influence  of  this  sphere  of  ac- 
tivity is  one  of  the  most  important  points  in  the  theory  of  electricity, 
because  no  other  shows  so  clearly  the  particular  laws  of  electric 
statics. 

396.  According  to  the  hypothesis  of  Symmer,  the  following  is  the 
order  of  the  phenomena  in  the  sphere  of  activity.     If  the  conductor 
of  the  machine  is  charged  with  vitreous  electricity,  the  insulated  con- 
ductor having  the  two  electricities  combined,  its  resinous  electricity 
is  attracted  by  the  vitreous  electricity  of  the  conductor  ;  it  is  not  by 
this  means  taken  away,  but  disguised,  so  that  its  influence  upon  the 
vitreous  electricity  of  the  conducting  body  is  diminished.     This  last 
is  therefore  free  to  a  certain  point,  and  becomes  the  more  so  as  the 
body  is  brought  nearer  the  conductor.     If  we  remove  the  body,  the 
repulsive  effect  which  the  electricity  of  the  conductor  produced  upon 
its  natural  vitreous  electricity,  is  weakened ;  consequently  the  two 
electricities  of  this  body  combine  more  fully,  and  the  effect  upon  its 
natural  electricity  becomes  less  sensible  j  finally,  when  the  body  is 
entirely  without  the  sphere  of  activity  of  the  conductor,  it  returns  to 
the  state  of  equilibrium  in  which  all  the  phenomena  disappear. 

But  if  we  touch  the  body  while  near  the  conductor,  we  take  from 
it  only  its  vitreous  electricity,  which  is  then  but  imperfectly  combin- 
ed ;  and  its  resinous  electricity  remains,  because  it  is  retained  and 
disguised  by  the  vitreous  electricity  of  the  conductor.  If  afterwards 
we  remove  the  body,  what  remains  of  its  natural  vitreous  electricity 
is  not  sufficient  to  disguise  its  resinous  electricity  ;  consequently  the 
latter  becomes  more  and  more  free,  and  thus  produces  its  accustom- 
ed effect. 

We  have  only  to  change  the  expressions  vitreous  and  resinous, 
and  this  explanation  will  apply  to  the  case  when  the  conductor  is 
electrified  resinously. 

We  see  that  these  phenomena  are  naturally  deduced  from  the 
hypothesis  of  Symmer. 

397.  To  explain  the  formation  of  the  sphere  of  activity,  accord- 
ing to  the  same  hypothesis,  it  is  only  necessary  to  admit  that  the 
two  electricities  act  upon  each  other  at  a  distance,  but  that  this  ac- 
tion has  no  other  influence,  except  to  diminish  their  reciprocal  activ- 
ity, and  cannot  take  away  either  from  the  bodies  in  which  they  are 
fixed. 


152  Electricity. 

If  one  of  the  two,  the  vitreous  electricity,  for  example,  is  accumu- 
lated in  a  body,  it  attracts  the  resinous  electricity  contained  in  the 
combination  of  the  two  electricities  of  the  surrounding  air  ;  at  the 
same  time  it  repels  the  vitreous  electricity.  By  this  double  influ- 
ence, it  diminishes  the  mutual  action  which  had  hitherto  rendered 
the  combination  without  effect.  Hence  the  vitreous  electricity  of 
the  nearest  stratum  of  air  becomes  almost  entirely  free,  and  pro- 
duces a  similar,  but  more  feeble  effect  upon  the  two  electricities  of 
the  surrounding  strata ;  and  this  influence  thus  propagates  itself 
from  stratum  to  stratum  to  a  greater  or  less  distance,  according  as 
the  force  of  the  vitreous  electricity  with  which  the  effect  began,  is 
more  or  less  intense.  According  to  this  explanation,  neither  of  the 
two  electricities,  any  where  in  the  sphere  of  activity,  is  in  its  natural 
state  ;  but  one  is  in  a  constrained  state  which  is  more  considerable 
in  proportion  as  it  is  nearer  the  body  actually  electrified. 

When  an  insulated  conductor  is  placed  in  the  sphere  of  activity, 
the  resinous  electricity  of  its  natural  state  is  combined,  to  a  certain 
degree,  with  the  vitreous  electricity  of  the  sphere  of  activity,  and 
consequently  its  vitreous  electricity  becomes  sensible  to  a  certain 
point. 

When,  on  the  contrary,  an  uninsulated  conductor  is  placed  in  this 
sphere  of  activity,  although  the  same  thing  takes  place,  the  effect  is 
different,  because  the  vitreous  electricity  escapes  through  the  con- 
ductor which  is  presented  to  it,  apd  there  remains  only  the  resinous 
electricity  in  a  combined  state. 

In  this  sense  we  are  to  understand  the  expression,  that  the  sphere 
of  activity  always  tends  to  excite  in  the  body  which  is  placed  in  it, 
an  electricity  opposite  to  its  own. 

398.  If  the  two  sides  of  a  thin  glass  plate  are  covered  with  tin 
foil  to  within  an  inch  or  two  round  the  border  of  the  plate,  so  that 
all  communication  between  the  two  metallic  coverings  is  cut  off, 
when  we  place  the  plate  in  such  a  manner  that  these  coverings  shall 
continue  insulated,  and  communicate  the  electricity  of  the  conductor 
to  the  upper  surface,  the  lower  surface  will  manifest  the  same  elec- 
tricity. If  we  take  the  electricity  from  the  upper  surface,  by  means 
of  a  point  directed  towards  it,  the  electricity  of  the  lower  surface  also 
disappears ;  yet  the  electricity,  in  these  circumstances,  escapes  with 
greater  difficulty  than  in  any  other  arrangement.  If,  on  the  contrary, 
we  take  away  the  electricity  of  the  lower  surface  by  contact,  and 


I 

Sphere  of  Activity.  183 

then  apply  a  point  to  the  upper  surface,  the  lower  surface  indicates 
an  increasing  but  opposite  electricity. 

399.  We  cannot  fail  to  perceive,  in  the  essential  circumstances  of 
ihis  experiment,  a  perfect  resemblance  to  that  of  article  394,  when 
we  suppose  the  electric  sphere  of  the  upper  surface  to  extend 
through  the  glass  to  the  lower  surface.     This  opinion  seems  to  be 
perfectly  confirmed  by  the  observation  of  article  382.     If  the  upper 
surface  is  charged  with  vitreous  electricity,  it  neutralizes  the  natural 
resinous  electricity  of  the  lower  surface,  and  then  the  vitreous  elec- 
tricity of  this  becomes  free.     If  we  take  the  vitreous  electricity  from 
the  upper  surface,  the  resinous  electricity  of  the  lower  surface  be- 
comes again  combined  with  the  vitreous  electricity  of  that  surface, 
and  the  \vhole  is  disguised.     But  if  we  first  touch  the  lower  surface, 
tve  take  away  its  free  vitreous  electricity ;  then  there  only  remains 
its  resinous  electricity,  which  is  neutralized  by  the  vitreous  electricity 
of  the  upper  surface,  as  long  as  this  remains  fixed ;  but  if  we  take 
this  vitreous  electricity  from  the  upper  surface  ;  the  resinous  electri- 
city of  the  lower  surface  becomes  free. 

400.  By  comparing  the  two  'experiments,  the  essential  conditions 
of  the  formation  of  a  sphere  of  activity,  are  determined  with  still 
more  exactness  by  the  following  enunciation. 

A  conducting  body  must  be  near  another  electrified  body,  and  be 
separated  from  it  by  a  non-conducting  medium.  We  shall  see 
hereafter  that  the  electrified  body  may  as  well  be  a  non-conductor 
as  a  conductor.  In  the  air,  the  distance  between  these  two  bodies 
may  be  very  considerable,  because  then  the  sphere  of  activity  ex- 
tends very  far.  But  if  the  non-conducting  medium  is  dense  and 
compact,  as  glass,  for  example,  the  sphere  of  activity  only  extends 
to  small  distances  ;  for  this  reason,  the  glass  employed  in  this  experi- 
ment must  not  be  too  thick.  In  these  circumstances,  the  conductor 
brought  near,  always  exhibits  the  phenomena  of  the  sphere  of  activ- 
ity, provided  the  electricity  be  not  accumulated  to  such  a  degree 
that  the  conductor  comes  within  the  striking  distance.  This  last 
case,  as  we  shall  see  hereafter,  may  take  place  spontaneously,  in 
consequence  of  a  very  great  charge,  even  when  the  separating  me- 
dium is  glass. 


134  Electricity. 


Accumulated  Electricity. 

401.  When  we  cause  the  inferior  surface  of  the  plate  to  commu- 
nicate with  the  ground,  and  then  electrify  the  superior,  this  last 
receives  a  much  greater  quantity  of  electricity  than  when  both  are 
insulated.     A  glass  plate  electrified   in   this  manner  is  said  to  be 
charged,  and  the  electricity  is  said  to  be  accumulated,  because  its 
effects  are  strikingly  distinguished  from  those  of  the  ordinary  electri- 
city hitherto  considered.     If  we  touch  only  the  lower  surface,  no 
effect  is  produced,  because  its  free  electricity  has  passed  into  the 
earth ;  if  we  touch  the  upper  surface  only,  we  receive  its  electricity, 
not  like  that  of  the  conductor,  by  a  single  spark,  but  by  several 
small  sparks,  producing  a  sharp  sensation,   and  succeeding  each 
other  rapidly.    Finally,  if  we  touch  both  surfaces  at  once,  we  receive 
the  whole  electricity  of  the  plate  by  one  strong  discharge,  which  not 
only  produces  a  painful  sensation  in  the  part  to  which  it  is  directed, 
but  also  in  both  arms,  and  especially  at  the  elbow  joints ;  it  is  called 
the  electric  shock,  and  is  never  produced  by  the  feeble  degrees  of 
electricity  obtained  in  the  ordinary  way.     This  discharge  of  the 
glass  plate  may  not  only  be  effected  by  the  hands,  but  by  any  other 
conducting  communication  between  the  metallic  surfaces.     If  we 
continue  to  charge  the  plate  the  electricity  always  accumulates,  till 
a  spontaneous  discharge  takes  place,  by  means  of  a  spark  from  the 
upper  surface,  which  traverses  the  glass  to  the  lower  surface,  and 
often  breaks  the  plate.     It  is  also  a  circumstance  worthy  of  remark, 
that  this  accumulated  electricity  does  not,  in  general,  act  so  strongly 
upon  the  electrometer  as  free  electricity.     We  may  be  convinced 
of  this  by  causing  a  quadrant  electrometer  to  communicate  with  the 
upper  surface  of  the  plate  while  it  is  charging,  or  what  amounts  to 
the  same,  by  attaching  it  to  the  conductor.* 

402.  It  is  truly  surprising  to  see  how  easily  all  these  phenomena 
are  explained  by  the  hypothesis  of  Symmer. 

If  the  conductor  furnishes  vitreous  electricity,  this  accumulates 
upon  the  upper  surface  of  the  plate,  until  its  sphere  of  activity  ex- 

*  For,  notwithstanding  the  great  quantity  of  electricity  accumu- 
lated, the  electrometer  will  only  indicate  a  feeble  tension,  on  ac- 
count of  the  attractive  action  of  the  opposite  electricity,  spread  over 
the  other  surface  of  the  plate. 


Accumulated  Electricity.  185 

tends  through  the  glass  to  the  lower  surface.  Then  the  lectricity 
of  the  upper  surface  neutralizes  the  resinous  electricity  of  the  lower 
surface  ;  but  as  each  combination  is  reciprocal,  it  is  itself  neutralized 
to  a  certain  degree,  and  thus  the  upper  surface  is  placed  to  a  cer- 
tain point  in  a  non-electric  state,  at  least  as  long  as  it  does  not  con- 
tain more  vitreous  electricity  than  it  might  have  received  without  this 
combination.  But  the  vitreous  electricity  becoming  free  on  the 
lower  surface,  passes  off  and  gives  place  to  the  new  combined  elec- 
tricity which  comes  from  the  ground.  This  is  decomposed  by  the 
vitreous  electricity  which  is  accumulated  on  the  upper  surface,  like 
that  which  existed  there  before,  and  it  will  be  easily  seen  that  this 
operation  will  continue  without  interruption,  so  long  as  the  upper  sur- 
face receives  an  excess  of  vitreous  electricity.  But  while  the  elec- 
tricity thus  accumulates,  the  striking  distance  extends  farther  into 
the  glass  ;  and,  if  it  attains  the  lower  surface,  a  spontaneous  discharge 
will  take  place. 

If  we  discharge  the  plate  before  this  happens,  all  the  electricity 
accumulated  on  the  upper  surface,  combines  instantaneously  with 
all  the  electricity  accumulated  on  the  lower  surface,  by  the  shortest 
course  which  is  offered  to  it,  and  produces  the  electric  shock  at  the 
moment  of  the  instantaneous  passage  through  the  body. 

The  diminution  of  the  electricity  upon  the  electrometer  arises 
from  this  ;  that  the  two  electricities  can  only  take  effect  in  the  free 
state,  and  on  the  plate  they  exist  in  a  certain  state  of  combination. 
This  combination,  however,  is  not  a  real  union  of  the  two  fluids, 
since  this  takes  place  afterward  by  the  discharge.  Each  of  the  elec- 
tricities adheres  to  the  surface  to  which  it  is  brought ;  but  in  their 
proximity  the  action  of  the  one  represses  that  of  the  other. 

403.  If  we  cover  the  upper  surface  with  a  substance  which  con- 
ducts electricity  badly,  for  example,  with  a  varnish  mixed  with  me- 
tallic powder,  the  plate  becomes  charged,  but  the  electricity  does 
not  diffuse  itself  tranquilly  over  the  surface  ;  on  the  contrary,  it  darts 
from  the  middle  towards  all  the  sides  in  serpentine  streaks.  If  we 
cause  the  lower  coating  to  communicate  with  the  edge  of  the 
upper  by  means  of  a  strip  of  tin  foil,  it  will  still  be  charged  ; 
but  when  the  electricity  is  accumulated  to  a  certain  degree  a  spon- 
taneous discharge  takes  place  through  the  strip  of  tin  foil,  which 
gives  beauty  to  the  phenomenon.  Such  a  plate  is  called  a  magic 
square. 

Elem.  24 


!S6  Ehctrkity. 

404.  As  the  form  of  the  glass  plate  for  this  experiment  is  entirely 
arbitrary,  we  ordinarily  use,  in  the  production  of  accumulated  elec- 
tricity, the  apparatus  called  the  Leydenjar.  The  arrangement  now 
considered  best  for  this  purpose  is  the  following.  We  coat  the 
interior  and  exterior  surfaces  of  a  glass  jar  with  tin  foil,  except 
an  inch  or  two  at  the  top,  which  is  usually  covered  with  sealing-wax 
dissolved  in  alcohol,  because  such  a  covering  insulates  better  than  the 
glass  alone.  The  interior  coating  is  charged  immediately  by  the 
conductor;  and  in  order  to  charge  it  more  conveniently,  a  small 
metallic  rod  is  inserted  into  the  jar,  reaching  three  or  four  inches 
beyond  the  mouth,  and  having  the  outer  extremity  terminated  with  a 
ball ;  the  lower  extremity  is  attached  to  a  round  plate  of  lead,  which 
adapts  itself  exactly  to  the  bottom  of  the  jar.  This  rod  is  supported 
at  the  top  of  the  jar,  by  passing  through  a  pasteboard  cover,  covered 
with  sealing-wax,  near  the  place  where  the  interior  coating  ends. 

In  order  to  charge  the  jar,  we  take  it  in  the  hand,  and  present 
the  ball  to  the  conductor ;  or  else  we  place  it  on  the  table,  and  cause 
the  ball  to  communicate  with  the  conductor  by  means  of  a  chain.* 

405.  As  it  would  be  dangerous  to  discharge  very  great  quantities 
with  the  hands,  we  employ  for  this  purpose  a  particular  instrument 
called  a  discharger.  It  consists  of  a  curved  metallic  rod,  rounded 
at  the  two  extremities,  and  terminated  by  two  balls ;  sometimes  one 
end  has  a  ring  and  the  other  a  ball.  We  place  one  end  in  contact 
with  the  exterior  coating  of  the  jar,  and  touch  the  ball  of  the  jar 
with  the  other.  In  this  way  the  whole  charge  passes  through  the 
metallic  arc,  without  creating  the  least  sensation  in  the  hands.  The 
discharger  is  still  more  convenient  when  it  consists  of  two  metallic 
arcs,  moveable  upon  an  insulating  handle  of  dry  wood  coated  with 
sealing-wax  ;  so  that  we  can  vary  at  pleasure  the  distance  between 
the  two  extremities ;  one  of  which  is  commonly  terminated  by  a 
ring,  and  the  other  by  a  ball. 


*  It  is  asserted  in  many  works,  that  thick  glass  is  less  liable  to 
break  than  thin.  But  this  is  not  true.  The  jars  made  by  Elckner, 
of  Berlin,  like  all  other  instruments  of  this  skilful  artist,  are  made 
with  great  care  ;  they  never  break  by  an  artificial  discharge,  and 
support  almost  all  spontaneous  discharges  ;  and  they  owe  this  supe- 
riority to  being  very  thin,  and  of  a  density  as  nearly  uniform  as  pos- 
sible. If  the  explanation  we  have  given  of  the  sphere  of  activity  b« 
correct,  this  fact  agrees  perfectly  with  the  theory. 


Accumulated  Electricity.  187 

406.  The  discovery  of  accumulated  electricity  was  made  in  the 
year  1745,  at  the  same  time,  by  two  observers ;  the  prebendary 
Kleist,  at  Cammin,  and  Muschenbroeck  at  Leyden.     Hence  the 
jar  is  called  the  Leyden  jar  or  jar  of  Kleist,  and  the  experiment 
itself  the  Leyden  experiment. 

407.  By  means  of  the  Leyden  jar,  a  variety  of  instructive  and 
amusing  experiments  may  be  performed.     With  respect  to  the  first 
we  shall  only  remark  what  follows. 

The  discharge  always  takes  place  when  we  establish  a  communi- 
cation between  the  two  coatings  of  the  jar,  whatever  be  the  extent  of 
this  communication.  We  may,  therefore,  give  the  electric  shock  to 
a  great  number  of  persons  at  the  same  time,  if  they  join  hands,  the 
first  in  the  series  touching  the  outer  coating  of  the  jar,  and  the  last 
the  ball  at  the  end  of  the  rod. 

If  electricity  be  left  to  pass  either  through  a  good  or  a  bad  con- 
ductor, it  takes  the  former  and  does  not  touch  the  latter.  For  this 
reason  we  may  hold  the  discharger  in  the  hand,  without  experienc- 
ing any  shock.  When,  however,  the  passage  through  a  bad  con- 
ductor is  much  the  shorter,  the  electricity  will  sometimes  follow  this 
instead  of  the  other. 

The  chain  of  communication  may  even  be  interrupted  at  one  or 
several  places  ;  and,  provided  the  distance  is  not  too  great,  the  dis- 
charge takes  place,  yet  with  this  difference,  that  at  each  interruption 
the  passage  is  attended  with  a  spark  which  gives  a  shock. 

408.  When  we  wish  to  augment  the  effect  of  the  Leyden  jar  as 
much  as  possible,  we  unite  several  jars  together,  and  thus  form  what 
is  called  the  electric  battery,  a  communication  taking  place  between 
all  the  exterior  coatings,  and  also  between  all  the  interior.    The  first 
is  easily  effected  by  placing  all  the  jars  on  the  same  sheet  of  tin 
foil ;  and  the  last  by  joining  all  the  rods  or  their  balls  with  a  metallic 
rod. 

This  apparatus  is  charged  from  the  machine,  by  causing  the  con- 
ductor to  communicate  with  the  interior  surface  of  one  of  the  jars  by 
means  of  a  wire  or  chain. 

We  cannot  determine,  in  general,  how  many  times  the  plate  must 
be  turned  to  obtain  a  complete  charge,  since  this  depends  not  only 
upon  the  size  of  the  machine,  but  also  upon  the  state  and  tempera- 
ture of  the  air. 

As  the  effects  of  a  charged  battery  are  such  as  to  render  it  neces- 
sary to  be  on  our  guard,  it  is  to  be  regretted  that  no  means  have 


jgg  Electricity. 

been  devised  for  determining  with  certainty  to  what  degree  the  bat- 
tery is  charged.  Under  these  circumstances  we  must  make  the 
best  use  of  the  imperfect  indications  which  are  in  our  power.  1 .  In 
each  case  we  should  count  how  many  times  we  have  turned  the 
plate,  in  order  to  obtain  a  rule  from  the  first  experiment  to  guide  u» 
in  the  next.  2.  The  quadrant  electrometer  should  always  be  placed 
upon  the  conductor  and  observed.  It  rises,  indeed,  much  more 
slowly  by  the  charge  of  a  battery  than  by  the  electricity  of  a  single 
conductor,  or  the  charge  of  a  single  jar  ;  but  we  may  deduce  from 
its  state  during  the  first  experiment  a  rule  for  those  which  follow ; 
and  we  may  also  observe,  to  a  certain  degree,  the  progress  of  the 
charge  in  the  first  experiment.  3.  Before  beginning  the  experiment, 
we  place  on  the  metallic  communications  which  unite  the  interior 
surfaces,  a  metallic  rod  terminated  at  the  two  extremities  by  a  ball 
about  i  of  an  inch  in  diameter,  and  prolonged  to  some  inches  be- 
yond the  battery.  From  time  to  time  we  apply  the  ball  of  an  insu- 
lated discharger  to  one  of  the  extremities  of  this  rod.  At  a  certain 
distance  a  spark  is  obtained,  and  this  distance  affords  a  tolerable 
indication  of  the  force  of  the  charge.  For  a  battery  of  20  or  30 
quart  jars  a  distance  of  half  an  inch  indicates  a  very  powerful  charge. 
4.  If  upon  observing  what  passes  in  the  battery  we  hear  a  crackling 
noise,  we  should  hasten  to  discharge  it,  for  this  indicates  either  that 
there  is  one  damaged  jar,  or  that  a  spontaneous  discharge  is  about 
to  take  place. 

We  should  carefully  avoid  a  spontaneous  discharge,  because  this 
almost  always  breaks  one  or  more  jars.  The  artificial  discharge  is 
made  in  the  same  manner  as  for  a  single  jar,  by  establishing  a  com- 
munication between  the  two  coatings. 

409.  The  effect  of  the  battery  increases  with  the  number  of  jars 
employed,  or  rather  with  the  extent  of  coated  surface. 

Regard  must  also  be  paid  to  the  force  of  the  electrical  machine 
employed.  The  more  feeble  it  is,  the  longer  is  the  time  required 
to  charge  the  battery,  and  the  greater  is  the  quantity  of  electricity 
lost  by  contact  with  the  air,  which  obstructs  the  charging. 

410.  The  universal  discharger  of  Henley  is  an  almost  essential  ap- 
pendage to  an  electric  battery.     It  is  constructed  as  follows.     Upon 
a  small  board  about  12  inches  in  length,  and  6  or  8  in  breadth,  are 
screwed  near  the  extremities,  two  insulating  columns  8  or  10  inches 
high.     Each  of  them  supports  a  metallic  rod  placed  transversely 
and  terminating  at  one  end  in  a  ring,  and  at  the  other  in  a  ball,  or  in 


Accumulated  Electricity.  181 } 

a  point,  when  the  ball  is  unscrewed.  These  metallic  rods  are  attach 
ed  to  the  column  by  means  of  a  socket  and  hinge,  in  such  a  mannei  . 
as  to  have  three  kinds  of  motion.     Each  rod  moves  forward  and 
backward   in  a  tube,  and  turns  horizontally  and  vertically.     Be- 
tween the  two  columns  is  a  small  table  of  dry  wood,  which  may  be 
raised,  depressed,  or  taken  away  at  pleasure.     Upon  the  small  table 
is  also  a  plate  of  the  same  size,  fitted  with  screws  so  as  to  form  a 
press. 

In  order  to  make  use  of  this  apparatus,  we  attach  one  of  the  ends 
of  a  metallic  chain  to  the  ring  of  one  of  the  transverse  rods,  and 
the  other  to  the  exterior  coating  of  the  battery.  We  put  the  body 
to  be  submitted  to  the  shock,  upon  the  small  table,  or  press  it  be- 
tween the  two  plates  ;  then  we  give  to  the  balls  of  the  two  metallic 
rods  a  suitable  situation  and  distance  with  respect  to  the  body  ;  we 
attach  the  ring  of  the  common  discharger  to  the  ring  of  the  second 
metallic  rod,  and  touch  with  the  ball  of  the  common  discharger, 
the  interior  coating  of  the  battery.  Then,  as  will  be  easily  seen,  the 
electric  spark  is  obliged  to  pass  through  the  body  which  is  placed 
between  the  two  balls  of  the  universal  discharger. 

411.  Among  the  innumerable  experiments  which  are  made  with 
an  electric  battery  we  shall  mention  only  the  following.  Birds  and 
other  small  animals  are  killed  instantaneously  by  the  discharge  of  a 
battery.  To  make  the  experiment  upon  larger  animals,  it  is  neces- 
sary to  use  much  caution.  Caterpillars  appear  to  form  an  exception, 
and  are  capable  of  sustaining  the  discharge  of  a  battery. 

The  spark  from  a  battery  passes  through  a  plate  of  thin  glass, 
with  a  great  noise,  but  without  shivering  it ;  it  only  makes  a  hole 
almost  imperceptible. 

It  pierces  through  several  folds  of  paper  or  pasteboard,  a  pack  of 
cards,  sheets  of  tin  or  lead  ;  and  it  is  remarkable  that  the  perfora- 
tions in  the  case  of  paper  have  a  sort  of  burr  projecting  each  way 
from  the  middle.  The  electric  spark  renders  red  hot,  melts,  or 
burns  fine  metallic  wires.  It  is  very  easy  to  make  this  experiment, 
even  with  small  batteries,  if  we  take  a  small  portion  of  a  very  fine 
wire,  for  example,  one  of  the  smallest  steel  chords  used  in  pianos. 

A  leaf  of  gold  or  silver,  pressed  between  two  plates  of  glass,  forms 
an  incrustation  on  the  glass  by  the  electric  spark.  A  leaf  which 
contains  alloy,  loses  by  the  operation  a  part  of  its  colour  in  several 
places,  which  is  the  effect  of  an  incipient  oxydation. 

If  we  apply  the  two  balls  of  Henley's  discharger  to  the  surface  of 


Electricity. 

at  5  or  6  inches  from  one  another,  the  discharge  takes  place 
a  loud  report.     If  we  hold  the  finger  in  the  water  during  the 
jarge  we  experience  a  certain  sensation.     Undoubtedly  the  va- 
r  above  the  water  favours  the  discharge. 

If  we  put  the  balls  of  the  discharger  in  water,  at  a  small  distance 
om  each  other,  the  spark  appears  in  the  water  between  the  balls, 
vhich  gives  the  water  a  singular  motion. 


CHAPTER  XXXV. 

Electrophorus  and  Condenser. 

412.  Alexander  Volta  of  Pavia,  has  enriched  the  electrical  appa- 
ratus with  several  very  remarkable  instruments,  among  which  are 
the  electrophorus  and  condenser.     It  is  the  characteristic  of  the  dis- 
coveries of  this  philosopher,  that  no  part  is  the  result  of  chance  ; 
the  whole  is  the  fruit  of  study,  and  of  the  application  of  theoretical 
principles. 

Electrophorus. 

413.  This  instrument  consists  of  a  circular  plate  of  tin,  surrounded 
by  a  border  slightly  raised.    .Its  size  is  very  variable.     Some  are 
only  a  few  inches  in  diameter,  and  others  are  2  or  3  feet.     The 
border  should  be  raised  in  proportion  to  the  size,  or  about  £  of  an 
inch  in  the  smallest,  and  an  inch  in  the  largest.     It  is  filled  with 
resin,  sealing-wax,  sulphur,  or  other  resinous  composition.     Care 
should  de  taken  that  the  surface  be  free  from  cracks  and  inequalities 
of  every  kind. 

The  third  and  last  essential  part  is  the  cover.  It  consists  of  a  cir- 
cular plate,  the  diameter  of  which  is  smaller  by  |  or  T\,  than  that 
of  the  part  above  described.  It  should  be  of  some  conducting  sub- 
stance, and  have  no  angle  or  prominence.  If  it  is  of  tin,  the  border 
should  be  rounded ;  but  then  it  is  difficult  to  preserve  it  perfectly 
plane,  and  without  inequalities  ;  for  which  reason  it  is  better  to  make 
it  to  consist  of  several  pieces  of  pasteboard  placed  one  upon  the 
other  and  covered  with  tin  foil.  We  suspend  it  by  three  silk  cords, 
like  the  scale  of  a  balance,  or  furnish  it  with  a  glass  handle.  Lastly, 
it  must  be  capable  of  being  removed  separately. 


Electrophorus.  191 

414.  We  excite  the  electricity  of  the  resinous  plate  by  rubbing  it 
with  cat-skin,  or  the  tail  of  a  fox,  perfectly  dry,  which  gives  it  the 
resinous  electricity. 

The  properties  which  distinguish  the  electrophorus  will  appear 
from  the  following  experiments  ; 

(1.)  If  we  place  the  cover  upon  the  electrified  resinous  plate,  the 
electricity  is  preserved  there  several  days,  and  even  weeks.  Hence 
the  name  electrophorus  or  bearer  of  electricity. 

(2.)  If  we  place  an  electrometer  upon  the  cover,  before  placing  it 
upon  the  resinous  plate,  and  then  bring  it  gradually  towards  the  plate, 
the  threads  of  the  electrometer  will  diverge  as  it  approaches.  The 
cover  is,  therefore,  electrified,  and  its  electricity  is  the  same  as  that 
of  the  resinous  plate.  But  if  we  remove  the  cover  without  touching 
it,  the  threads  collapse,  and  in  proportion  to  the  distance  ;  so  that 
when  it  is  without  the  sphere  of  activity  of  the  plate,  all  signs  of  elec- 
tricity disappear.* 

(3.)  If  we  replace  the  cover,  and  touch  it  before  taking  it  away, 
or  what  is  still  better,  if  we  touch  at  the  same  time,  and  with  the 
same  hand,  the  metallic  part  of  the  plate  and  the  cover,  the  finger 
which  touches  the  cover  receives  a  slight  spark,  and  the  threads  of 
the  electrometer  collapse,  so  that  the  cover  will  seem  no  longer 
to  possess  any  electricity.  But  if  we  now  remove  it  by  the  insulating 
handle,  the  threads  of  the  electrometer  will  diverge  and  remain  at  a 
certain  distance  when  it  is  without  the  sphere  of  activity  of  the  plate  ; 
the  cover  is  therefore  electrified,  but  with  a  different  electricity  from 
that  of  the  plate.  If  we  again  replace  the  cover,  the  threads  col- 
lapse, and  all  signs  of  electricity  disappear ;  but  if  we  touch  the 

*  When  we  make  this  experiment,  the  cover  must  not  be  suffered 
to  remain  too  long  in  contact  with  the  resinous  plate,  or  even  in  its 
sphere  of  activity  ;  for,  as  the  natural  electricity  of  the  cover  is 
decomposed,  and  the  vitreous  part  only  is  retained  by  the  attraction 
of  the  plate,  while  the  resinous  is  repelled,  the  latter  has  a  tendency 
to  escape.  It  is  this  which  causes  the  threads  to  diverge.  Now 
as  the  surrounding  air  never  produces  a  perfect  insulation,  a  part 
of  the  electricity  escapes  in  this  way  ;  and  though  the  effect  is  incon- 
siderable during  a  short  interval  of  time,  yet  as  it  is  constantly  re- 
peated, the  plate  will  in  time  be  discharged  of  its  resinous  electricity, 
precisely  as  if  we  had  touched  it  ;  and  we  can  perceive  its  succes- 
sive diminutions  by  the  gradual  collapsing  of  the  threads. 


192  Electricity. 

cover  before  replacing  it,  we  receive  a  considerable  spark,  by  wbicli 
all  its  electricity  is  taken  away. 

(4.)  We  can  repeat  this  experiment  as  often  as  we  please,  and 
alternately  take  sparks  from  the  cover,  when  removed  and  when 
replaced,  without  diminishing  the  electricity  of  the  plate. 

415.  After  \vhat  has  been  said  in  the  preceding  chapter,  respect- 
ing the  sphere  of  activity,  these  phenomena  require  little  explanation. 
The  only  new  circumstance  here  exhibited,  is,  that  there  is  no  true 
communication,  but  simply  a  separation  of  electricity,  when  we  place 
the  cover  on  the  plate.     We  have  already  seen  that  the  form  of 
bodies  has  a  great  influence  upon  the  communication  of  electricity  ; 
and  that  the  communication  is  more  difficult,  in  proportion  as  the 
body  offers  fewer  points  or  angles.     The  electrophorus  then  teaches 
us  a  new  law  of  communication  ;  between  two  plane  surfaces,  one 
of  which  is  of  a  conducting  nature  and  the  other  a  non-conductor, 
there  can  be  no  communication. 

416.  The  electrophorus  may  supply  the  place  of  the  electrical 
machine  in  a  great  number  of  cases ;  for  when  it  is  once  electrified, 
it  is,  if  we  may  so  speak,  an  inexhaustible  source  of  electricity.    By 
means  of  it  we  can  even  charge  Leyden  jars,  and  give  them  either 
kind  of  electricity  at  pleasure.     For  this  purpose,  we  place  two  jars 
near  the  electrophorus,  and  cause  the  exterior  coating  of  one  to 
communicate  with  the  metallic  cover.     This  jar  takes  sparks  from 
the  cover  when  it  is  on  the  resinous  plate,  and  the  other  jar  takes 
sparks  from  it  when  it  is  removed.     The  first  is,  therefore,  charged 
with  resinous  electricity,  and  the  second  with  vitreous.     But  it  takes 
a  long  time  to  obtain  powerful  charges,  unless  the  electrophorus  is 
very  large. 

417.  We  shall  here  mention  an  easy  method   of  accumulating 
electricity  on  the  plate.     We  charge  a  jar  with  electricity  by  mccins 
of  the  electrophorus  or  machine,  and  place  it  on  the  plate ;  this 
being  done,  we  take  it  by  the  knob,  and  move  it  over  the  electropho- 
rus ;  in  this  manner  all  the  vitreous  electricity  of  the  jar  passes  grad- 
ually into  the  hand,  and  the  plate  as  gradually  takes  all  the  resinous 
electricity,  which  it  retains  combined,  and  thus  its  electricity  is  aug- 
mented. 

418.  Among  the  experiments  to  be  performed  only  with  the  elec- 
trophorus,  there  is  one  which  consists  in  producing  certain  appear- 
ances called  Lichtenberg's  figures.     We  charge  two  jars  with  dif- 
ferent electricities.     We  take  each  of  them  by  the  outer  coating. 


Condenser.  193 

and  draw  figures  on  the  resinous  plate  with  the  knobs,  having  pre- 
viously removed  all  other  electricity  from  the  plate,  by  rubbing  it 
and  then  wiping  it  with  a  linen  cloth.  This  being  done,  we  spread 
some  fine  powder  over  the  resinous  plate,  as  sulphur  1,  red  lead,  &£. 
and  the  figures  traced  by  one  and  the  other  of  these  electricities,  are 
easily  distinguished  by  means  of  the  particles  of  powder,  which  ar- 
range themselves  about  the  outlines  of  the  figures. 


Condenser. 

419.  Great  pains  have  lately  been  taken  to  examine  the  feeble 
degrees  of  electricity  which   are  manifested    in  many   cases.      It 
is  important,  therefore,  to  be  acquainted  with  the  instruments  neces- 
sary for  such  investigations.     The  principle  one,  besides  very  sensi- 
ble electrometers,  is  the  condenser  of  Volta,  by  means  of  which  the 
most  feeble  quantities  of  electricity  may  be  detected  and  observed. 

420.  The  construction  of  the  condenser  may  be  varied  in  differ- 
ent ways,  but  it  is  always  extremely  simple.     The  essential  parts 
are  the  cover  and  the  base.     The  cover  is  disposed  like  that  of  the 
electrophorus,  only  it  is  commonly  smaller,  being  from  2  to  5  inches 
in  diameter.     It  is  convenient  to  have  it  of  metal,  the  lower  surface 
being  polished.     The  base  is  a  plate  of  a  little  larger  diameter  ;  it 
should  be  made  of  some  non-conducting  substance,  or  if  we  use  a 
conductor,  it  should  be  covered  with  some  substance   which  will 
not  allow  the  electricity  to  penetrate  it.     Commonly  it  is  a  polished 
disc  covered  with  taffeta  or  a  thin  layer  of  varnish.     We  might  also 
use  dry  marble,  or  dry  wood  varnished,  &c. ;  or,  since  the  air  is  a 
bad  conductor,  we  may  place  three  small  glass  plates  upon  a  table 
as  a  support,  and  put  the  cover  above  them  ;  thus  making  the  stra- 
tum of  air  below  answer  for  a  base. 

421.  In  the  electrophorus  the  resinous  plate  is  electrified  ;  in  the 
condenser  we  communicate  the  feeble  quantity  of  electricity,  which 
is  to  be  examined,  directly  to  the  cover  while  it  is  on  the  base.     So 
long  as  it  remains  on  the  base,  it  shows  scarcely  any  signs  of  elec- 
tricity ;  but  if  we  remove  it,  it  produces  a  sensible  effect  upon  the 
electrometer,  and  even  gives  sparks.     We  might  make  the  experi- 
ment with  the  small  quantity  of  electricity  which  a  jar  retains  after  a 
discharge. 

Elem.  25 


194  Electricity. 

422.  From  what  has  already  been  said,  the  theory  of  the  conden- 
ser presents  no  difficult}'*.  There  is  no  communication  between  the 
cover  and  the  base  ;  consequently  the  electricity  communicated  to 
the  cover  forms  a  sphere  of  activity.  If  we  give  the  vitreous  elec- 
tricity to  the  cover,  it  neutralizes  to  a  certain  degree,  the  natural 
resinous  electricity  of  the  base,  and  is,  of  course,  itself  neutralized  to 
the  same  degree.  In  this  way,  as  with  the  jar,  the  cover  is  capable 
of  accumulating  more  electricity  ;  it  therefore  absorbs  all  the  elec- 
tricity of  the  bodies  which  are  presented  to  it ;  but  this  electricity  is 
entirely  or  nearly  disguised,  as  long  as  the  cover  is  placed  upon  the 
base  j  but  when  removed  beyond  the  sphere  of  activity  of  the  base, 
the  electricity  becomes  free  and  manifests  itself  in  the  usual  way. 

It  will  be  readily  seen  how  this  instrument,  so  useful  for  measuring 
small  quantities  of  electricity,  may  be  applied  in  other  cases. 


CHAPTER  XXXVI. 

Electricity  Excited  by  other  Means  beside  Friction. 

423.  Several  philosophers  of  the  last  century,  and  particularly 
Nollet,  Winkler,  and  Franklin,  conceived,  about  the  same  time,  the 
idea  that  lightning  was  an  electrical  phenomenon.  But  the  cele- 
brated Franklin  not  only  had  the  incontestable  merit  of  deciding  the 
question  by  actual  experiment ;  but  he  acquired  a  lasting  fame  also 
by  the  invention  of  lightning  rods.  A  detailed  exposition  of  the 
theory  of  thunder  and  lightning  belongs  rather  to  physical  geography 
than  to  mechanical  philosophy  ;  still  we  have  introduced  the  subject 
here,  because  it  furnishes  evidence,  in  a  striking  manner,  that  there 
is  in  nature  the  means  of  exciting  a  powerful  electricity,  of  which  we 
have  yet  perhaps  no  knowledge  ;  for  we  have  not  the  slightest  rea- 
son to  believe  that  the  electricity  which  manifests  itself  in  thunder 
and  lightning,  is  produced  by  the  friction  of  air  against  air,  or  of  air 
against  the  vapour  of  water.  The  following  circumstance  demands 
some  attention.  During  a  shower,  the  atmosphere  being  filled  with 
the  vapour  of  water  and  drops  of  rain,  has  a  communication  of  a  con- 
ducting nature  with  the  ground,  by  which  a  very  considerable  quan- 
tity of  electricity  is  insensibly  drawn  from  the  cloud.  But,  as  very 
powerful  discharges  take  place  in  the  mean  time,  and  are  often  con- 


Electricity  Excited  by  other  Means  beside  Friction.        195 

tinued  lor  several  hours,  we  are  compelled  to  admit,  that  there  is 
some  constant  operation  in  the  cloud  itself,  by  which  so  great  a  quan- 
tity of  electricity  becomes  free,  that  the  conducting  power  of  the  air 
cannot  convey  it  away,  or  even  sensibly  diminish  it.  As  to  the  man- 
ner in  which  this  phenomenon  takes  place,  we  are  unable  to  offer 
any  satisfactory  theory. 

424.  Friction  is  undoubtedly  the  most  active  means  of  exciting 
electricity  ;  and  it  is  certain  that  electricity  is  always  excited  when- 
ever two  bodies  are  rubbed  together,  especially  if  they  are  not  ho- 
mogeneous. It  may  be  that  this  electricity  does  not  produce  a  sen- 
sible effect,  either  on  account  of  its  feeble  intensity,  or  because  it  is 
immediately  conveyed  off  by  the  conducting  media  which  are  pre- 
sented to  it.  But  we  are  now  acquainted  with  other  means  of 
exciting  electricity,  besides  friction,  though  only  in  feeble  degrees. 

Among  these  means  we  should  first  mention  the  great  influence 
which  heat  and  cold  have  upon  electrical  phenomena.  They  change 
the  conductibility  of  bodies.  Glass,  heated  to  redness,  becomes  a 
conductor;  and  ice,  in  a  state  of  extreme  cold  [below —  13  of 
Fahrenheit]  becomes  a  non-conductor.  Siliceous  earth,  heated  in  a 
crucible,  shows  an  electric  attraction  for  the  sides  of  the  vessel.  The 
electrical  phenomena  presented  by  a  heated  tourmaline,  are  exceed- 
ingly curious. 

A  vast  field  for  research  is  here  open  to  chemists ;  for  there  is 
reason  to  believe  that  in  each  chemical  combination,  changes  are 
produced  in  the  electric  state  of  bodies.  Indeed  traces  of  electri- 
city are  discovered  when  water  passes  to  a  state  of  vapour ;  when 
charcoal  is  consumed  ;  when  sulphur,  wax,  and  resin  are  melted ; 
and,  according  to  the  acute  observation  of  Lavoisier  and  Laplace, 
when  iron  is  dissolved  in  sulphuric  acid,  &c. 

It  is  exceedingly  desirable  that  chemists  should  complete  the  in- 
vestigation of  this  subject ;  for  it  may  lead  to  the  most  interesting  re- 
sults. It  is  obvious  how  necessary  it  is  in  such  inquiries  to  have 
instruments  capable  of  rendering  sensible  the  slightest  degrees  of 
electricity. 


196  Electricity. 


Galvanism. 

425.  The  most  important  discovery  that  has  been  made  in  our 
time  on  this  subject,  is  that  of  the  developement  of  electricity  by  the 
simple  contact  of  two  metals,  one  of  which  takes  the  vitreous,  and 
the  other  resinous  electricity.     This  discovery  has  given  rise  to 
new  and  remarkable  results.     Means  have  been  discovered  of  con- 
siderably augmenting  the  electricity  obtained  in  this  way,  and  it  is 
thus  found  to  produce  effects  which  are  absolutely  peculiar  to  it ; 
so  that   some  philosophers   still  question  its  perfect  identity  with 
common  electricity.*     All  that  belongs  to  this  new  discovery  has 
received  the  name  of  galvanism,  from  Galvani,  a  philosopher  of  Bo- 
logna, who  first  observed  the  phenomenon  which  has  led  to  these 
researches ;  yet  we  owe  to  the  sagacity  of  Volta,  all  the  most  im- 
portant discoveries  connected  with  the  subject. 

426.  In  1791,  Galvani  accidentally  perceived  that  the  thigh  of  a 
frog,  separated  from  the  body  and  skinned,  experienced  contrac- 
tions, when  made  to  communicate  with  two  metals,  one  in  contact 
with  the  nerve,  and  the  other  with  the  muscle.     He  afterwards 
found  this  phenomenon  to  take  place  equally  in  every  part  of  the 
animal,  but  that  the  irritability  of  the  muscle  necessary  to  produce 
it,  continued  only  a  short  time  after  death.     These  experiments 
were  soon  repeated  with  various  modifications  throughout  Europe. 
We  shall  state  the  most  interesting  facts  which  have  been  made 
known  by  Galvani  and  others. 

(1.)  The  experiment  succeeds  with  all  the  metals,  and  even  with 
some  other  bodies,  as  charcoal,  plumbago,  &c. ;  but  it  is  best  to  em- 
ploy zinc  in  connexion  with  gold,  silver,  or  copper. 

(2.)  Instead  of  two  metals  we  may  employ  a  kind  of  galvanic 
chain  of  several  bodies,  one  end  terminating  in  a  nerve,  and  the 
other  in  a  muscle.  The  effect  takes  place  as  soon  as  the  circuit  is 
completed.  But  it  appears  that  all  bodies  are  not  equally  adapted 
to  this  purpose ;  and  that  the  distinction  between  conductors  and 
non-conductors  of  electricity  obtains  here  also. 

*  This  might  have  been  true  when  the  author  wrote ;  but  since 
Volta  proposed  his  ingenious  theory,  there  has  been  but  one  opinion 
among  enlighted  inquirers  on  the  subject. 


Galvanism.  197 

(3.)  It  is  not  necessary  that  one  end  of  the  chain  should  termi- 
nate in  a  nerve  and  the  other  in  a  muscle  ;  both  may  be  terminated 
by  a  nerve  or  by  the  muscular  fibres  which  communicate  with  the 
nerve. 

(4.)  The  presence  of  water  appears  to  be  an  essential  condition 
of  this  phenomenon ;  for  when  the  parts  of  the  animal  placed  in 
contact,  are  not  moist,  there  is  no  effect,  or  at  most  a  very  feeble 
one. 

(5.)  The  experiment  may  be  performed  upon  animals  of  what- 
ever kind,  and  even  upon  the  separate  parts  of  the  human  body. 
But  the  irritability  continues  longer  after  death  in  cold-blooded  ani- 
mals, than  in  the  warm-blooded. 

(6.)  The  contact  of  two  metals  also  produces  striking  effects  upon 
the  living  body.  If  we  place  two  pieces  of  different  metals  upon 
one  or  two  incisions  made  in  any  part  of  the  body,  we  experience  a 
sharp  pain,  when  the  two  metals  are  brought  into  contact.*  If  we 
put  a  piece  of  zinc  under  the  tongue,  and  a  piece  of  silver  above  ity 
and  bring  the  pieces  of  metal  into  contact,  we  experience  a  deci- 
dedly acid  taste.  If  we  change  the  order  of  the  pieces  of  metal  the 
taste  is  different,  and  according  to  some,  alcaline.  If  we  place  one 
piece  of  metal  against  the  internal  angle  of  the  eye,  and  the  other 
between  the  lower  lip  and  the  jaw,  we  see  at  the  moment  of  contact, 
a  flash  nearly  resembling  distant  lightning.  Some  also  pretend  they 
perceive  a  very  subtile  difference  in  the  light,  when  the  order  of  the 
metals  is  changed. 

427.  At  first,  philosophers  differed  much  in  the  explanation  they 
gave  of  these  phenomena.  Some  thought  they  had  discovered  a 
new  natural  force,  which  acted  only  upon  the  animal  organization, 
and  which,  therefore,  ought  to  be  termed  animal  electricity.  Many 
considered  these  as  purely  electrical  phenomena,  but  disagreed  in 
their  explanation.  Galvani  supposed  that  in  the  living  state,  the 
interior  of  the  nerves  contained  vitreous  electricity ;  that  the  muscles 
or  exterior  envelope  of  the  nerves  contained  resinous  electricity,  and 
that  these  experiments  were  analogous  to  the  discharge  of  a  Leyden 
jar.  Volta,  on  the  contrary,  had  observed  that  the  simple  contact  of 


*  This  experiment  was  performed  by  M.  Humboldt ;  applying 
pieces  of  gold  to  two  blisters  made  on  his  shoulder,  he  experienred 
all  the  effects  here  described  ;  and  this  discovery  suggested  several 
important  physiological  remarks. 


198  Electricity. 

two  metals,  excited  in  both  a  feeble  degree  of  electricity,  so  that 
resinous  electricity  could  be  detected  in  the  one,  and  vitreous  in  the 
other  ;  and  he  maintained  that  this  property,  taken  in  connexion  with 
the  well  known  susceptibility  of  the  nerve  to  the  action  of  the  feeblest 
degree  of  electricity,  furnished  the  true  explanation  of  these  phe- 
nomena. This  opinion  appears  to  be  confirmed  by  all  the  experi- 
ments that  have  been  since  made. 


Voltaic  Pile,  or  Galvanic  Battery. 

428.  Volta  was  led  by  reasoning  alone,  and  not  by  conjecture,  to 
the  discovery  of  a  method,  by  which  this  kind  of  electricity  might  be 
wonderfully  augmented  ;  this  is  called  the  Voltaic  pile.    In  order  to 
construct  it,  we  place  silver  and  zinc,  or  copper  and  zinc  plates, 
alternately  one  above  the  other,  and  separate  each  pair  by  a  piece 
of  cloth  moistened  with  water  or  a  saline  solution.     The  order  must 
continue  the  same  throughout  the  series ;  thus  silver,  zinc,  water, 
&c.,  so  that  the  two  extremities  of  the  pile  shall  terminate  with  differ- 
ent metals.     Each  extremity  takes  the  name  of  silver  or  zinc  pole, 
according  to  the  metal  which  terminates  it.     In  order  to  observe  the 
effects  in  a  satisfactory  manner,  the  pile  should  consist  of  at  least  50 
pairs.     The  plates  may  be  about  the  size  of  a  crown  or  dollar.    We 
shall  speak  in  particular  of  the  effect  of  larger  plates. 

In  general,  the  column  is  disposed  in  such  a  manner  as  to  be  com- 
pletely insulated.  We  often  begin  and  end  the  pile  by  double  plates, 
between  which  is  interposed  a  thin  strip  of  brass,  leaving  on  one 
side  a  small  projection,  to  which  wires  may  be  attached  in  perform- 
ing the  experiments. 

429.  The  most  remarkable  phenomena  exhibited  by  such  a  pile 
are  the  following  : 

(1.)  If  we  attach  wires  to  the  extremities  of  the  pile,  and  take 
one  of  these  in  each  hand,  we  experience  a  painful  sensation  which 
is  repeated  continually,  as  long  as  the  communication  is  kept  up. 
This  effect  is  more  energetic  when  the  hands  are  moistened  ;  and 
still  more  so,  when  we  take  a  piece  of  wet  metal  in  each  hand,  the 
wires  communicating  with  these ;  or  when  the  extremities  of  the 
wires  are  immersed  in  a  vessel  of  water,  and  we  touch  the  water 
with  both  hands. 


Voltak  Pile,  or  Galvanic  Battery.  199 

The  effect  of  the  pile  may  be  produced  upon  any  part  of  the  body, 
and  through  a  number  of  persons  forming  an  arc  of  communica- 
tion. 

(2.)  The  luminous  appearance  mentioned  in  article  426,  may 
easily  be  produced  and  varied  by  means  of  the  pile.  For  this  pur- 
pose we  have  only  to  take  one  of  the  wires  in  the  moist  hand,  and 
bring  the  other  to  the  eye  also  moistened,  or  to  the  tongue.  In  the 
last  case  we  experience,  moreover,  an  extremely  acrid  taste.  All 
the  phenomena  of  articles  425,  426,  become  more  conspicuous 
and  striking  by  means  of  the  pile. 

(3.)  When  we  cause  the  wires  to  communicate  with  two  very 
delicate  electrometers,  they  manifest  feeble,  but  unequivocal  signs  of 
electricity.  The  zinc  pole  always  exhibits  vitreous  electricity,  and 
the  silver  or  copper  pole  resinous.  These  electricities  may  be  ob- 
served still  better  by  means  of  a  small  condenser. 

With  this  apparatus  we  can  charge  small  jars,  trace  the  figures  of 
Lichtenberg  on  the  electrophorus,  &c. 

(4.)  If  we  attach  an  iron  wire  to  one  pole  and  touch  the  other 
pole  with  the  same  wire,  we  perceive  a  spark.*  The  experiment  is 
more  certain  when  we  envelope  the  extremity  of  the  iron  wire  in  a 
thin  gold  leaf.  This  leaf  is  consumed  at  the  place  through  which 
the  spark  passes.  By  means  of  gold  leaf  we  can  also  inflame  de- 
tonating gas,  phosphorus,  sulphur,  &#. 

(5.)  The  most  important  experiment  performed  with  the  pile  be- 
longs to  chemistry.  But  it  is  so  remarkable  that  we  cannot  omit 
noticing  it.  We  refer  to  the  decomposition  of  water.  To  effect 
this,  we  fill  a  glass  tube  with  distilled  water  and  close  both  extremi- 
ties with  cork  stoppers.  The  wires  attached  to  the  two  poles  are 
made  to  pass  through  these  stoppers  and  terminate  in  the  water  at  the 
distance  of  a  few  lines  from  each  other.  The  ends  of  the  wires  are 
commonly  sharpened,  but  this  is  not  essential.  The  wires  may  be  of 
silver  or  any  grosser  metal.  In  the  latter  case  we  observe  the  follow- 
ing phenomena.  The  extremity  of  the  wire  attached  to  the  silver 
or  copper  pole,  disengages  bubbles  from  the  water,  which  accumu- 
late in  the  upper  part  of  the  tube.  Having  collected  a  sufficient 
quantity  of  the  gas  for  examination,  we  find  it  to  be  hydrogen,  one 

*  If  we  attach  two  very  fine  wires  to  the  poles,  and  bring  their 
extremities  gradually  into  contact,  an  attraction  takes  place,  which 
retains  them  together. 


200  Electricity. 

of  the  constituent  principles  of  water.  The  extremity  of  the  wire 
connected  with  the  zinc  pole,  is  covered  with  the  oxyde  of  the  metal 
of  which  the  wire  is  formed,  which  proves  that  oxygen  has  been 
disengaged  at  this  wire.  Thus  we  find  the  two  constituent  princi- 
ples of  water. 

When  the  two  wires  are  of  platina  or  pure  gold,  gas  is  disengaged 
from  each ;  hydrogen,  as  before,  from  that  connected  with  the  copper 
pole,  and  oxygen  from  that  connected  with  the  zinc.  In  this  case  we 
employ  a  recurved  tube  in  the  form  of  the  letter  F,  for  the  purpose 
of  collecting  and  examining  the  two  gases  separately.  This  decom- 
position may  also  be  effected  by  common  electricity,  but  not  so  con- 
veniently nor  so  abundantly. 

(6.)  In  general,  the  electricity  of  the  pile  is  much  more  efficient 
in  its  chemical  effects,  than  in  those  which  are  mechanical.  In  the 
pile  itself,  we  not  only  perceive  a  decomposition  of  the  water  with 
which  the  interposed  cloths  are  wet,  but  also,  when  a  saline  solution 
is  used,  we  perceive  a  decomposition  of  the  salt,  which  strongly 
attacks  and  oxydates  the  metallic  plates  between  which  the  cloths 
are  placed.  On  this  account  many  have  thought  that  the  electricity 
of  the  pile  is  to  be  attributed  rather  to  this  chemical  action,  than  to 
the  contact  of  the  metals ;  but  the  principles  established  by  Volta, 
as  well  as  the  experiments  themselves,  are  opposed  to  such  a  suppo- 
sition. 

430.  From  some  late  experiments,  it  appears  that  the  intensity  of 
some  of  its  effects  is  in  proportion  to  the  height  of  the  pile  or  the 
number  of  pairs,  while  that  of  others  depends  upon  the  size  of  the 
plates. 

The  effects  produced  upon  the  bodies  of  animals,  vary  with  the 
number  of  pairs  ;  but  the  greater  or  less  size  of  the  plates  seems  to 
have  little  or  no  influence.  On  the  contrary,  the  chemical  effects 
are  much  more  powerful  when  the  plates  are  6  or  8  inches  in  diam- 
eter, than  when  they  are  only  2  or  3.  There  is  reason  to  believe 
that  no  metal  is  capable  of  resisting  the  heat  of  the  electric  pile. 
Silver,  gold,  and  platina,  melt  and  become  oxydated  with  a  beauti- 
ful blue  light,  that  is,  they  burn.  But  the  metal  must  be  reduced  to 
very  thin  leaves  before  it  is  employed  for  this  purpose. 


Relations  of  Electricity  and  Galvanism.  201 


Relations  of  Electricity  and  Galvanism. 

431.  It  is  remarkable  that  among  the  many  strong  resemblances 
between  electricity  and  galvanism,  we  nowhere  find  a  perfect  ac- 
cordance. The  sensation  produced  by  the  pile,  is  very  different 
from  that  produced  by  the  jar.  With  small  plates  we  obtain  only  a 
feeble  spark ;  with  large  plates,  the  chemical  effects  of  the  spark  far 
exceed  those  of  common  electricity.  The  phenomena  of  attrac- 
tion and  repulsion,  as  well  as  the  charging  of  jars,  are  produced  by 
the  pile  with  great  difficulty  ;  whereas  water  is  decomposed  with 
much  greater  facility  by  means  of  the  pile,  than  by  common  electri- 
city. Insulation,  without  which  most  of  the  experiments  made  with 
common  electricity  do  not  succeed,  appears  to  be  of  little  importance 
in  most  of  the  experiments  which  are  performed  by  means  of  the 
pile  ;  yet  this  condition  becomes  essential  when  we  wish  to  produce 
an  effect  on  the  electrometer,  or  charge  a  jar  or  condenser.  The 
presence  of  water  is  entirely  unnecessary  in  most  electrical  experi- 
ments ;  but  becomes  an  essential  condition  for  nearly  all  galvanic  ex- 
periments. 

Nevertheless,  as  all  these  differences  result  rather  from  diversities 
of  intensity,  than  from  actual  anomalies  in  what  constitutes  the  phe- 
nomena, we  cannot  doubt  the  identity  of  the  force  which  is  exerted 
in  the  two  cases.  Indeed,  we  may  easily  conceive  that  there  must 
be  a  great  difference  in  the  effects,  when  we  reflect  that  almost  all 
the  phenomena  of  common  electricity  are  produced  by  an  instanta- 
neous motion  of  the  electric  matter  j  whereas  the  phenomena  of 
galvanism  are  produced  simply  by  a  constant  current  of  this  matter. 


Addition. 

432.  To  complete  the  view  of  galvanism  presented  by  the  author, 
and  to  show  its  resemblance  in  all  respects  to  common  electricity,  I 
have  thought  it  proper  to  annex  a  report  made  to  the  French  Na- 
tional Institute,  on  the  subject  of  Volta's  experiments. 

The  first  galvanic  phenomena  consisted  in  muscular  contractions 
excited  by  the  contact  of  a  metallic  arc.  Galvani  and  many  others 
regarded  them  as  the  result  of  a  particular  kind  of  electricity,  inher- 
ent in  animals.  Volta  first  showed  that  the  animal  arc  made  use  of 

Elem.  26 


202  Electricity. 

in  these  experiments,  served  only  to  receive  and  manifest  ihe  galvanic 
influence.  He  considered  the  muscular  irritation,  which  was  first 
thought  to  be,  the  most  important  part  of  the  phenomenon,  as  nothing 
more  than  the  effect  of  the  electric  action  produced  by  the  mutual 
contact  of  the  metals,  of  which  the  exciting  arc  was  formed.  This 
opinion  which  found  many  advocates  and  many  opposers,  led  to  a 
variety  of  experiments  intended  to  support  or  refute  it ;  and  the 
effect  was  such  as  is  always  witnessed  in  the  infancy  of  discoveries. 
A  multitude  of  apparent  anomalies  presented  themselves,  which  were 
absolutely  inexplicable,  on  account  of  the  delicate  circumstances  ac- 
companying them,  the  influence  of  which  was  not  yet  known. 

433.  Such  was  the  state  of  this  branch  of  science  when  the  com- 
mittee on  galvanism  made  their  first  report  to  the  Institute.     Their 
aim  was  to  determine  accurately  the  conditions  necessary  for  devel- 
oping and  modifying  the  galvanic  effects.     They  did  riot  attempt  to 
explain  them ;  but  confined  themselves  to  a  mere  statement  of  facts, 
in  the  order  which  seemed  most  proper.  At  this  time  the  researches 
by  which  Volta  had  endeavoured  to  connect  with  his  first  discovery 
all  the  phenomena  which  galvanism  presented,  were  unknown  in 
France.     This  distinguished  philosopher  has  since  recognised  many 
other  facts  which  he  has  combined  together  in  an  ingenious  theory. 
If  there  still  remains  something  to  be  done  in  order  to  determine  with 
exactness  the  laws  of  this  singular  action,  and  subject  them  to  a  rig- 
orous calculation,  the  principal  facts  which  are  to  serve  as  a  basis, 
seem  to  be  firmly  established. 

434.  The  principal  fact  from  which  all  the  rest  are  derived>  is  the 
following.     If  two  different  metals,  insulated,  and  having  only  their 
natural  quantity  of  electricity,  are  brought  into  contact,  and  then 
withdrawn,  we  find  them  in  different  electric  states,  one  being  posi- 
tive and  the  other  negative. 

This  difference,  which  is  very  small  at  each  contact,  being  suc- 
cessively accumulated  in  a  condenser,  becomes  sufficient  to  produce 
a  sensible  divergence  in  the  threads  of  the  electrometer.  The  ac- 
tion is  not  exerted  at  a  distance,  except  when  different  metals  are  in 
contact ;  it  continues  during  the  contact,  but  its  intensity  is  not  the 
ysame  for  all  metals. 

It  is  sufficient  to  take,  for  an  example,  copper  and  zinc.  By  their 
mutual  contact  the  copper  is  electrified  negatively,  and  the  zinc  posi- 
tively. 


Theory  of  the  Voltaic  Pile.  203 

After  having  proved  the  developement  of  metallic  electricity,  inde- 
pendently of  every  moist  conductor,  Volta  introduced  these  conduc- 
tors. 

435.  If  we  form  a  metallic  plate,  consisting  of  two  pieces,  one  of 
zinc  and  one  of  copper,  soldered  end  to  end,  and  taking  the  zinc  ex- 
tremity between  the  fingers,  apply  the  other  extremity,  of  copper,  to 
the  upper  plate  of  the  condenser,  which  is  likewise  of  copper,  this 
will  be  charged  negatively  j  as  is  evident  from  the  preceding  experi- 
ment. 

If,  on  the  contrary,  we  take  the  copper  end  between  the  fingers, 
and  touch  the  upper  or  copper  plate  of  the  condenser,  with  the  zinc 
extremity,  the  plate  of  the  condenser  on  being  separated,  will  be 
found  not  to  have  acquired  any  electricity,  although  the  lower  plate 
communicates  with  the  common  reservoir. 

436.  But  if  we  place  between  the  upper  plate  and  the  zinc  ex- 
tremity a  paper  moistened  with  pure  water,  or  any  other  moist  con- 
ductor, the  condenser  becomes  charged  with  positive  electricity.    If, 
under  these  circumstances,  we  touch  the  plate  with  the  copper  ex- 
tremity, it  still  becomes  charged,  but  negatively.     These  facts  admit 
of  no  dispute.     The  following  is  Volta's  explanation  of  them. 

The  metals  (says  he)  and  probably  all  bodies,  exert  a  reciprocal 
action  upon  their  respective  electricities,  at  the  moment  of  contact. 
When  we  take  the  metallic  plate  by  its  copper  extremity,  a  part  of 
its  electric  fluid  passes  into  the  other  extremity,  which  is  of  zinc  ; 
but  if  this  zinc  is  in  immediate  contact  with  the  condenser,  which  is 
also  of  copper,  the  latter  tends  to  discharge  its  fluid  with  an  equal 
force,  and  the  zinc  cannot  transmit  any  to  it ;  the  plate  must,  there- 
fore, be  in  its  natural  state  after  contact.  •  If,  on  the  contrary,  we 
place  a  moistened  paper  between  the  zinc  plate  and  the  copper  plate 
of  the  condenser,  the  moving  property  of  the  electricity,  which  ex- 
ists only  in  the  case  of  contact,  is  destroyed  between  these  metals ; 
the  water  which  appears  to  possess  this  property  only  in  a  very 
feeble  degree,  compared  with  the  metals,  does  not  prevent  the  trans- 
mission of  the  fluid  from  the  zinc  to  the  condenser,  and  this  becomes 
charged  positively.  Lastly,  when  we  touch  the  condenser  with  the 
copper  extremity,  the  interposed  moist  paper,  the  action  of  which  is 
very  feeble,  does  not  prevent  the  plate  of  the  condenser  from  trans- 
mating  a  part  of  its  positive  electricity  into  the  zinc  extremity,  and 
thus,  upon  destroying  the  contact,  the  condenser  is  negatively  charged. 


^04  Electricity. 

437.  According  to  this  theory  it  is  easy  to  explain  the  voltaic  pile. 
For  the  sake  of  greater  simplicity,  suppose  it  placed  on  an  insulator; 
and  let  unity  represent  the  excess  of  electricity,  which  a  zinc  piece 
must  have  over  a  copper  one  in  immediate  contact  with  it.*  If  the 
pile  is  composed  of  only  two  pieces,  the  lower  one  of  copper,  and 
the  other  of  zinc,  the  electric  state  of  the  first  will  be  represented  hy 
—  i,  and  that  of  the  second  by  -f-  !• 

If  we  add  a  third  piece,  which  must  be  of  copper,  it  will  be 
necessary  in  order  to  effect  a  transmission  of  the  fluid,  to  separate 
it  from  the  zinc  piece,  by  a  piece  of  moistened  pasteboard,  then  it 
will  be  in  the  same  electric  state  with  the  last ;  at  lt-;;st  if  we  leave  out 
of  consideration  the  proper  action  of  the  water,  which  appears  to  be 
very  feeble,  and  perhaps  also  a  slight  resistance  which  the  water,  as 
an  imperfect  conductor,  may  oppose  to  the  transmission.  The  ap- 
paratus being  insulated,  the  excess  of  the  upper  piece  can  only  be 
acquired  at  the  expense  of  the  copper  piece  which  is  below.  Then 
the  respective  states  of  these  pieces  will  no  longer  be  as  before. 

For  the  lower  piece  which  is  of  copper,  we  shall  have  —  f. 

For  the  second  winch  touches  it,  and  which  is  of  zinc,  —  f-  -\-  1 , 

««•  +  i; 

The  third  which  is  of  copper,  and  separated  from  the  preced- 
ing, will  have  the  same  quantity  of  electricity  ;  that  is,  -j-  ]  ;  and 
the  sum  of  the  quantities  of  electricity  lost  by  the  fira  piece,  and 
acquired  by  the  two  others,  will  be  equal  to  zero,  as  in  the  preceding 
case  of  two  pieces.  If  we  add  a  fourth  piece  which  will  be  of  zinc, 
it  must  have  an  unit  more  than  that  of  thu  copper,  which  is  immrdi- 
ately  under  it;  and  as  this  excess  can  only  be  acquired  at  the  ex- 
pense of  the  lower  pieces,  the  pile  being  insulated,  we  shull  have, 
for  the  lower  piece  which  is  of  copper,  —  1  ;  for  the  second,  which 

*  The  quantities  of  electricity  accumulated  in  a  body  beyond  its 
natural  state,  other  things  being  the  same,  are  proportional  to  tho 
repulsive  force  with  which  the  particles  of  the  fluid  tend  to  separate 
from  each  other,  or  to  repel  a  now  particle  which  we  endeavour  to 
add  to  them.  This  repulsive  forco,  which  in  free  bodies,  is  counter- 
acted by  the  resistance  of  the  air,  constitutes  what  we  call  the  f»i- 
sion  of  the  fluid  ;  this  tension  is  not  proportional  to  the  divergence 
of  the  straws  in  Volta's  electrometer,  or  of  tho  balls  in  that  of  Saus- 
sure;  it  can  be  exactly  measured  only  by  means  of  tho  electric  bal- 
ance. 


Theory  of  the  Voltaic  Pile.  205 

is  of  zinc  and  in  contact  with  the  first,  0  ;  that  is,  it  will  be  in  its  nat- 
ural state. 

For  the  third  piece,  which  is  of  copper,  and  which  is  separated 
from  the  preceding,  we  shall  have  0  ;  this  also  will  be  in  its  natural 
state.  For  the  fourth  piece,  which  is  of  zinc,  and  in  contact  with 
the  preceding,  we  shall  have  -f-  !• 

By  pursuing  the  same  reasoning  we  can  find  the  electric  states  of 
each  piece  of  the  pile,  supposing  it  insulated  and  formed  of  any 
number  of  elements.  The  quantities  of  electricity  will  increase  i'or 
each  of  them,  from  the  base  to  the  summit  of  the  column,  in  an 
arithmetical  progression,  the  sum  of  which  will  be  zero. 

4o8.  If,  for  greater  simplicity,  we  suppose  the  number  of  elements 
even,  it  is  easy  to  prove  by  a  simple  calculation,  that  the  lower  piece 
which  is  of  copper,  and  the  upper  piece  which  is  of  zinc,  must  be 
equally  electrified,  one  positively,  the  other  negatively  ;  and  the  same 
is  true  of  any  pieces  taken  at  equal  distances  from  the  extremities  oi 
the  pile. 

Before  passing  from  the  positive  to  the  negative,  the  electricities 
will  become  nothing  ;  and  there  will  always  be  two  pieces,  one  of 
zinc  and  the  other  of  copper,  which  will  be  in  their  natural  state  ; 
they  will  be  at  the  middle  of  the  pile  ;  this  we  have  seen,  for  exam- 
ple, in  the  case  of  four  pieces. 

Suppose  now  that  we  establish  the  communication  between  the 
lower  piece  and  the  common  reservoir  ;  it  is  evident  that  this  piece, 
which  is  negatively  electrified,  will  tend  to  recover  from  the  ground 
what  it  has  lost ;  but  its  electric  state  cannot  change  without  changing 
that  of  the  pieces  above  it,  since  the  electrical  interval  between  two 
successive  pieces  must  always  be  the  same  when  in  a  state  of  equili- 
brium. It  follows,  that  all  the  negative  quantities  of  the  lower 
half  of  the  pile,  will  be  neutralized  at  the  expense  of  the  common 
reservoir.  Then  the  state  of  the  pile  will  be  as  follows  ; 

The  lower  piece  which  is  of  copper,  will  have  the  electricity  of 
the  ground,  which  we  shall  call  zero. 

The  second  piece  which  is  of  zinc,  and  in  immediate  contact  with 
the  preceding,  will  have  -f-  1 . 

The  third  which  is  of  copper,  and  separated  from  the  last  by 
moistened  paper,  will  have  the  same  electricity,  namely,  -f-  1. 

The  fourth  which  is  of  zinc,  and  in  contact  with  the  preceding,  will 
have  -f  2. 

Thus  the  quantities  of  electricity  will  increase  upward  in  arithmeti- 
cal progression. 


206  Electricity. 

439.  Then,  if  we  touch  with  one  hand  the  summit  of  the  pile,  and 
with  the  other  the  base,  these  excesses  of  electricity  will  be  dis- 
charged through  the  organs  into  the  common  reservoir,  and  will  ex- 
cite a  sensation  so  much  the  stronger,   as  this    loss  repairs  itself 
at  the  expense  of  the  ground  ;  and  there  must  result  an  electric  cur- 
rent, the  rapidity  of  which  being  greater  in  the  interior  of  the  pile, 
than  in  the  organs  which  are  imperfect  conductors,  will  enable  the 
lower  part  of  the  pile  to  recover  a  degree  of  tension,  approaching 
that  which  it  had  in  the  state  of  equilibrium. 

The  communication  being  always  established  with  the  common 
reservoir,  if  we  put  the  summit  of  the  pile  in  contact  with  the  upper 
plate  of  a  condenser,  whose  base  communicates  with  the  ground, 
the  electricity  which  is  found  at  this  extremity  of  the  pile,  in  a  very 
feeble  degree  of  tension,  will  pass  into  the  condenser,  where  the  ten- 
sion may  be  regarded  as  nothing  ;  but  the  pile  not  being  insulated, 
this  loss  will  be  repaired  at  the  expense  of  the  common  reservoir  ; 
the  new  quantities  of  electricity,  recovered  by  the  upper  plate,  will 
pass  into  the  condenser,  like  the  preceding,  and  will  accumulate 
there  to  such  a  degree,  that  by  separating  the  collector  plate  we  may 
obtain  from  it  sensible  electrometrical  signs,  and  even  sparks.  As  to 
the  limit  of  this  accumulation,  it  evidently  depends  upon  the  thick- 
ness of  the  small  stratum  of  gum  lac,  which  separates  the  two  plates 
of  the  condenser  ;  for  in  consequence  of  this  thickness,  the  electri- 
city accumulated  in  the  collector  plate,  being  able  to  act  only  at  a 
distance  upon  that  of  the  lower  plate,  is  always  greater  than  that 
which  puts  it  in  equilibrium  in  this  last ;  and  hence  results,  in  the 
collector  plate,  a  slight  tension,  which  here  has  for  its  limit,  the 
tension  existing  in  the  upper  part  of  the  pile. 

In  like  manner  as  the  electricity  of  the  column  accumulates  in  the 
condenser,  it  will  also  accumulate  in  the  interior  of  a  Leyden  jar, 
whose  exterior  surface  communicates  with  the  common  reservoir ; 
and,  since  the  pile  continually  recharges  itself  at  the  expense  of  this 
same  reservoir,  the  jar  will  be  equally  charged,  whatever  be  its  capa- 
city. But  its  interior  tension  can  never  exceed  that  which  exists  at 
the  summit  of  the  pile.  If  we  then  remove  the  jar,  it  will  produce 
the  sensation,  corresponding  to  this  degree  of  tension ;  and  this  is 
confirmed  by  experiment. 

440.  Such  must  be  the  state  of  things,  if  we  neglect,  as  very 
small,  the  proper  action  of  the  water  upon  the  metals,  and  suppose, 
1.  That  the  transmission  of  the  fluid  takes  place  from  one  couple  to 


Theory  of  the  Voltaic  Pile.  207 

another  in  the  insulated  pile,  through  the  pieces  of  moistened  paste- 
board which  separate  them,  even  when  there  exists  no  other  com- 
munication between  the  two  extremities  of  the  column."  2.  That 
the  excess  of  electricity  which  the  zinc  takes  from  the  copper,  is 
constant  for  these  two  metals,  whether  they  are  in  their  natural  state 
or  not. 

Volta  established  the  first  proposition  by  an  experiment  already 
referred  to,  in  which  the  condenser  becomes  charged,  when  we 
touch  the  collector  plate,  covered  with  moistened  paper,  with  the 
copper  extremity  of  the  metallic  plate,  holding  the  zinc  extremity 
between  the  fingers. 

The  second  proposition  is  the  most  simple  that  can  be  imagined. 
M.  Coulomb  made  a  series  of  very  delicate  experiments  to  verify 
it,  and  it  appeared  to  him  exact.  I  have  also  myself  arrived  at  the 
»ame  results. 

441.  Hitherto,  for  the  sake  of  clearness,  we  have  supposed  the 
pile  to  be  formed  of  copper  and  zinc.     The  same  theory  would 
equally  apply  to  any  two  metals ;  and  the  effects  of  the  different 
arrangements  would  depend  upon  the  differences  of  electricity  estab- 
lished between  them  at  the  moment  of  contact. 

What  we  have  said  extends  equally  to  all  other  bodies  between 
which  there  exists  an  analogous  action.  Thus,  although  this  action 
appears  in  general  to  be  very  feeble  between  liquids  and  metals,  yet 
there  are  some  liquids,  as  the  alkaline  sulphurs,  between  which  and 
metals,  the  action  is  very  sensible.  Accordingly  the  English  have 
substituted  these  for  one  of  the  metallic  elements  of  the  pile  ;  and 
they  were  employed  at  a  still  earlier  date  by  M.  Pfaff  in  his  experi- 
ments. 

442.  Volta  discovered  between  metallic  substances  a  very  re- 
markable relation,  which  renders  it  impossible  to  construct  a  pile 
with  them  alone.     We  shall  state  it  after  his  own  manner,  having 
never  had  occasion  to  verify  it.     If  we  arrange  the  metals  in  the  fol- 
lowing order,  silver,  copper,  iron,  tin,  lead,  zinc,  each  will  become 
positive  by  contact  with  that  which  precedes  it,  and  negative  with 
that  which  follows  it ;  the  electricity  will,  therefore,  pass  from  the 
silver  to  the  copper,  from  the  copper  to  the  iron,  and  so  on. 

Now  the  property  in  question  consists  in  this ;  that  the  moving 
force  from  the  silver  to  the  zinc,  is  equal  to  the  sum  of  the  moving 
forces  of  the  metals  which  are  comprehended  between  them  in  the 
series  ;  whence  it  follows  that  if  we  put  them  in  contact  in  this  order 


208  Electricity. 

or  any  other  we  please,  the  extreme  metals  will  always  be  in  the 
same  state  as  if  they  immediately  touched  each  other.  Consequently, 
if  we  suppose  any  number  of  elements  thus  disposed,  the  extremes 
of  which  are  silver  and  zinc,  for  example,  we  should  have  the  same 
results  as  if  we  employed  only  these  two  metals  ;  that  is,  there  would 
be  no  effect,  or  it  would  be  the  same  with  that  which  a  single  cle- 
ment would  produce. 

443.  As  yet  it  appears  that  the  preceding  property  extends  to  all 
solid   bodies  ;  but   it  does    not  subsist  between  them  and    liquids. 
Hence  it  is  that  we  have  succeeded  in  the  construction  of  the  pile, 
by  the  intervention  of  liquids.    Hence,  too,  results  the  division  which 
Volta  has  made  of  conductors  into  two  classes  ;  the  first  comprising 
solids,  the  second  liquids.     No  pile  has  yet  been  constructed  with- 
out a  suitable  mixture  of  these  two  classes.     It  has  not  been  found 
possible  to  form  it  with  the  first  alone  ;  and  we  do  not  yet  know 
enough  of  the  mutual  action  of  liquids,  to  say  whether  it  is  the  same 
with  respect  to  them  or  not. 

444.  We  have  supposed  that  the  pieces  of  moistened  pasteboard, 
placed  between  the  elements  of  the  pile,  had  irnbihr-d  pure  water. 
If  we  employ,  instead  of  water,  a  saline  solution,  the  effect  becomes 
very  much  more  powerful ;  but  the  tension  indicated  by  the  elec- 
trometer does  not  appear  to  be  augmented,  at  least  in  the  same  ratio. 
Volta  has  established  this  fact  by  means  of  the  crown  of  cups,  by 
filling  them  successively  with  pure  and  acidulated  water. 

He  concludes  from  this  experiment,  that  the  acids  and  saline  solu- 
tions favour  the  action  of  the  pile,  chiefly  by  increasing  the  conduct- 
ing power  of  the  water,  with  which  the  pasteboard  is  moistened. 
As  to  the  oxydation,  he  regards  it  as  an  effect  which  establishes  a 
closer  contact  between  the  elements  of  the  pile,  and  thus  contributes 
to  render  its  action  more  sustained  and  energetic. 

I  have  since  verified  this  opinion  by  a  series  of  accurate  experi- 
ments, and  have  found  it  to  be  perfectly  correct.  Whatever  be  the 
substance  interposed  as  a  humid  body,  provided  there  is  the  same 
number  of  pairs,  the  condenser  is  charged  to  the  same  degree  ;  only 
it  requires  more  or  less  time,  according  as  the  substance  interposed 
has  a  greater  or  less  conducting  power. 

Such  is  a  brief  sketch  of  Volta's  theory  of  galvanic  electricity. 
His  aim  was  to  reduce  all  the  phenomena  to  one,  the  existence  of 
which  is  now  well  established  ;  namely,  the  developement  of  elec- 
tricity by  the  mutual  contact  of  metals.  It  appears  to  be  proved  by 


Addition.  209 

his  experiments,  that  the  particular  fluid  to  which,  for  some  time, 
the  muscular  contractions  and  the  phenomena  of  the  pile,  were  attri- 
buted, is  nothing  but  common  electricity,  put  in  motion  by  a  cause 
of  which  we  only  see  the  effects,  without  knowing  any  thing  of  its 
nature. 

445.  Such  is  the  fate  of  the  sciences,  that  thp  most  brilliant  dis- 
coveries only  open  a  field  for  new  researches.    After  having  discov- 
ered and  estimated,  if  we  may  so  say,  by  approximation,  the  mutual 
action  of  the  metallic  elements,  it  remains,  in  order  to  determine  it  in 
a  rigorous  manner,  to  see  whether  it  is  constant  in  the  same  metals, 
or  whether  it  varies  with  the  quantities  of  electricity  which  they  con- 
tain, and  with  their  temperature.     It  is  necessary  to  determine  with 
the  same  precision,  the  proper  action  exerted  by  the  liquids  upon  one 
another,  and  upon  the  metals.     When  this  is  done  we  shall  be  able 
to  proceed  in  our  calculations  upon  exact  data,  and  thus  ascend  to 
the  true  law  which  governs  the  distribution  and  motions  of  electri- 
city in  the  Voltaic  pile,  and  complete  the  explanation  of  all  the  phe- 
nomena which  this  apparatus  presents.  But  these  delicate  researches 
require  the  use  of  the  most  sensible  and  accurate  instruments  which 
have  yet  been  invented,  for  measuring  the  force  of  the  electric  fluid. 
Lastly,  it  remains  to  examine  the  chemical  effects  of  the  voltaic  cur- 
rent, its  action  upon  the  animal  economy,  and  its  connexion  with  the 
electricity  of  minerals  and  fishes  ;  these  researches  cannot  fail,  from 
the  facts  already  known,  to  be  very  important. 

446.  When  a  science,  already  in  an  improved  state,  has  received 
an  important  accession,  new  relations  are  established  between  the 
branches  which  compose  it.     It  is  then  pleasant  to  go  back  to  the 
beginning  and  retrace  the  steps  by  which  it  has  advanced.     If  we 
look  to  the  origin  of  this  science,  we  find  it,  in  the  beginning  of  the 
last  century,  reduced  to  the  simple  phenomena  of  attraction  and  re- 
pulsion.   Dufay  first  ascertained  the  constant  laws  to  which  these  are 
subject,  and  explained  their  apparent  contradictions.     His  discovery 
of  the  two  electricities,  vitreous  and  resinous,  was  the  foundation  of 
the  science ;  and  Franklin,  by  presenting  it  under  a  new  point  of 
view,  prepared  the  way  for  a  theory,  to  which  all  the  phenomena, 
even  that  of  the  Leyden  jar,  could  be  reduced.     Epinus  perfected 
this  theory  by  subjecting  it  to  the  calculus,  and  by  the  aid  of  analy- 
sis arrived  at  those  results  which  Volta  so  happily  employed  in 
the  condenser  and  electrophorus.     The  exact  law  of  electric  attrac- 
t/em. 27 


210  Electricity. 

tions  and  repulsions  was  still  wanting.     Coulomb  proved  that  it  is 
the  same  as  that  of  gravity  which  governs  the  celestial  motions. 

At  last  the  phenomena  of  galvanism  presented  themselves,  appa- 
rently differing  from  all  those  hitherto  observed.  In  order  to  explain 
them,  recourse  was  first  had  to  a  particular  fluid  j  but  by  a  course 
of  ingenious  experiments,  conducted  with  sagacity,  Volta  proposed 
to  reduce  them  all  to  one  single  cause,  the  developement  of  metallic 
electricity,  made  them  subservient  to  the  construction  of  an  appara- 
tus which  enables  us  to  increase  their  force  at  pleasure,  and  by  his 
results  connected  them  with  the  most  important  phenomena  of  chem- 
istry and  the  animal  economy.  [For  more  minute  information  upon 
this  subject,  the  reader  is  referred  to  the  Cambridge  Course  of  Na- 
tural Philosophy,  vol.  ii.] 


SECTION  VII. 

MAGNETISM. 


CHAPTER  XXXVII. 

General  Properties  of  the  Magnet. 

447.  AMONG  the  different  kinds  of  iron  ore  that  are  found  in  a  nat- 
ural state,  there  is  one  in  particular  which  possesses  the  surprising 
property  of  attracting  iron  by  an  invisible  force.  It  is  true  that  all  the 
fragments  of  this  kind  of  ore  do  not  possess  this  property  to  the  same 
degree ;  but  it  is  found  in  most  of  them,  and  even  in  considerable 
masses.  These  are  what  we  call  natural  magnets.  We  can  com- 
municate this  property  to  iron  and  steel,  and  thus  produce  artificial 
magnets.  It  was  formerly  thought  that  the  magnetic  force  belonged 
exclusively  to  iron  ;  but  it  is  now  found  to  be  common  to  two  other 
metals,  nickel  and  cobalt ;  and  its  energy  is  proportional  to  the  purity 
of  these  substances.  Yet,  as  it  is  difficult  to  obtain  considerable  masses 
of  these  substances  in  a  state  of  great  purity,  all  we  know  of  their 
magnetic  properties  is,  that  they  are  attracted  by  a  magnet.  But  it 
would  be  interesting  to  inquire,  whether,  like  iron,  they  are  capable 
of  receiving  and  communicating  magnetism.* 

*  I  have  made  this  experiment  upon  fragments  of  nickel,  prepared 
at  Berlin,  and  given  me  by  M.  Berthollet.  I  formed  them  into 
needles  by  laminating  them,  and  then  examined  their  magnetic  force 
by  means  of  oscillations,  a  method  which  will  be  explained  hereafter. 
I  thus  found  that  the  magnetic  force  of  nickel  was  about  %  that  of 
iron  of  the  same  weight.  It  has,  like  iron,  the  property  of  retaining 
and  communicating  the  magnetic  force.  But  these  results  are  not 
absolute ;  since  they  must  vary  with  the  coercive  force  given  to  the 
nickel  by  hammering. 


212  Magnetism. 

Relation  between  the  Magnet  and  Unmagnetized  Iron. 

448.  Iron  in  the  metallic  state  and  in  that  of  black  oxyde,  attaches 
itself  to  the  magnet  with  considerable  force.     In  the  last  case,  how- 
ever, it  must  not  be  strongly  oxydated.     This  force  is  measured  by 
the  weight  of  iron  which  the  magnet  is  capable  of  raising.     It  dots 
not  depend  upon  the  size  of  the  magnet ;  for  there  are  large  mag- 
nets which  have  very  little  force,  and  small  magnets  which  have  a 
very  great  force.     Sometimes  they  will  support  ten  times  their  o\vn 
weight.  Lastly,  experiment  shows  that  the  force  of  the  same  magnet 
varies  with  the  positions  in  which  it  acts. 

449.  The  magnetic  force  does  not  manifest  itself  with  equal  inten- 
sity at  all  the  points  of  the  surface  of  a  magnet.     Ordinarily 

are  two  points  of  this  surface  where  the  action  is  strongest ;  some- 
times, but  rarely,  there  are  more.  These  places  are  called  the 
poles  of  the  magnet.  They  are  determined  by  putting  the  n»:«;pet 
in  iron  filings.  The  filings  attach  themselves  much  more  sir  ;;!> 
about  the  poles,  than  any  where  else.  We  can  also  discover  them 
by  means  of  a  piece  of  very  fine  iron  wire.  At  the  poles  it  attaches 
itself  to  the  magnet  by  one  of  its  extremities,  keeps  the  other  at  the 
greatest  distance  ;  and  thus  remains  perpendicular  to  the  surface  of 
the  magnet ;  at  any  other  point  it  takes  an  oblique  position  directed 
towards  the  nearest  pole.  At  points  equally  distant  from  the  two 
poles,  it  applies  itself  longitudinally  to  the  surface. 

450.  When  the  two  poles  are  capable  of  acting  at  the  same  time 
upon  the  opposite  extremities  of  a  piece  of  iron,  the  magnetic  at- 
traction is  augmented.     For  this  reason,  we  often  give  to  artificial 
magnets  the  form  of  a  horse-shoe,  of  which  the  two  extremities  are 
the  poles.    We  apply  to  these  extremities  a  piece  of  soft  iron,  and 
load  it  with  as  much  iron  as  the  magnet  is  capable  of  supporting. 

451.  The  magnetic  force  is  exerted,  not  only  in  the  case  of 
a  contact,  but  also  at  a  distance.     A  common  magnet  will  raise 
filings  without  touching  them.      This   force  decreases  with  the 
distance  ;  but  in  a  magnet  of  an  irregular  form,  the  law  of  this  de- 
crease appears  to  be  very  complicated ;  and  although  it  is  easy  to 
determine  the  intensity  of  the  magnetic  force  for  each  distance,  by 
means  of  a  balance,  the  magnet  being  placed  in  it  and  counter- 
balanced,  yet  experiment  shows  that  the   law  of  decrease   is  so 
modified  by  the  form,  magnitude,  and  position  of  the  two  bodies, 


m 

Properties  of  the  Magnet.  2 1 3 

that  it  becomes  very  difficult  to  determine  with  exactness  the  proper 
law  and  the  influence  of  modifying  circumstances. 

452.  If  we  place  a  magnet  under  a  pane  of  glass,  wood,  paste- 
board, or  any  other  substance,  except  iron,  and  cover  the  surface 
with  iron  filings  j  upon  being  agitated  the  filings  will  arrange  them- 
selves in  a  certain  order,  and  form  curved  lines  extending  from  one 
pole  to  the  other,  so  that  we  can  easily  distinguish  these  points. 

This  experiment  indicates  that  the  magnetic  force  exerts  itself 
through  all  bodies,  except  iron,  which  according  to  the  manner  in 
which  it  is  placed,  increases  or  diminishes  the  magnetic  effect.  We 
shall  find  still  stronger  reasons  hereafter,  for  believing  that  this  force 
is  not  at  all  weakened  by  the  interposition  of  material  bodies.  On 
account  of  this  singular  property,  it  is  easy  to  conceal  either  the  mag- 
net or  the  iron  upon  which  it  acts.  And  it  is  in  this  way  that  the 
numerous  instruments  are  contrived,  which  are  used  in  the  tricks  of 
jugglers. 

453.  We  may  preserve  all  the  force  of  a  magnet,  and  even  some- 
times increase  it  by  taking  care  to  load  it  with  as  great  a  weight  as  it 
is  capable  of  supporting.     It  is  also  very  useful  to  give  the  poles 
their  natural  situation.     If  we  leave  the  magnet  without  being  thus 
loaded,  its  force  gradually  diminishes.     Very  small  magnets  may  be 
preserved  by  iron  filings.     Rust  weakens  the  magnetic  virtue  j  and 
magnets  strongly  heated  lose  entirely  their  magnetic  properties.     It 
has  also  been  observed  that  a  fall,  a  blow  with  a  stone,  or  an  elec- 
rical  discharge,  sometimes  impairs  the  magnetic  power. 


Properties  of  the  Magnet. 

454.  If  by  any  means  whatever,  we  place  a  magnet  in  such  a 
manner  that  it  can  move  freely  in  a  horizontal  direction,  it  always 
takes  such  a  position  that  one  of  its  poles  is  directed  towards  the 
north  and  the  other  towards  the  south.  For  this  reason  we  call  one 
of  these  the  north  pole  and  the  other  the  south. 

The  observation  of  this  property  led  to  the  invention  of  the  mari- 
ner's compass,  which  is  nothing  more  than  a  needle  of  magnetized 
steel,  placed  upon  a  pivot,  and  fitted  to  move  freely  in  a  horizontal  di- 
rection. It  is  hardly  possible  to  estimate  the  utility  of  this  simple  in- 
strument. Its  in  venter  is  not  known,  and  even  the  time  of  its  inven- 
tion is  uncertain,  though  it  is  generally  supposed  to  be  between  the 


214  Magnetism. 

12th  and  14th  century.     The  ancients,  although  unacquainted  with 
it,  had  noticed  the  attractive  property  of  the  magnet. 

However  little  connexion  there  appears  to  be,  at  first  view,  be- 
tween the  attractive  power  of  the  magnet  and  its  polarity,  we  shall 
see  hereafter  that  the  polarity  is  simply  the  effect  of  the  magnetic 
attraction  of  the  globe. 


The  Reciprocal  Action  of  Magnets. 

455.  Two  magnets  mutually  attract  each  other  at  determinate 
points,  and  this  attraction  is  even  greater  than  that  which  exists  be- 
tween the  magnet  and  iron.     In  other  points  they  repel  each  other  ; 
and  by  means  of  two  magnetized  needles,  or  a  magnet  and  a  mag- 
netized needle,  we  easily  verify  the  following  law. 

Poles  of  opposite  names  attract  each  otlier  ;  those  of  the  same  name 
repel  each  other. 

This  law  furnishes,  a  convenient  method  of  finding  the  poles  of  a 
magnet. 

456.  As  a  magnet  of  considerable  force  acts  at  a  great  distance 
upon  a  good  magnetized  needle,  we  can  easily  satisfy  ourselves  by 
experiment,  that  the  interposition  of  bodies  does  not  diminish  the 
magnetic  power. 

Communication  of  Magnetism. 

457.  We  can  communicate  a  sensible  magnetic  force  to  a  small 
piece  of  iron,  by  passing  it  several  times  across  the  pole  of  a  magnet. 
Only  we  must  take  care  to  pass  it  always  in  the  same  direction  ;  for, 
by  changing  the  direction,  we  diminish  the  magnetism  already  com- 
municated.    One  of  the  best  methods  of  performing  this  operation  is 
the  following.    Let  n  s  (fig.  56)  be  a  bar  of  uumaguetized  iron,  and 
JVS  a  magnet,  of  which  JV  is  the  north  pole  and  -S  the  south  pole. 
We  place  the  magnet  upon  the  iron  as  indicated  in  the  figure  ;  so 
that  the  north  pole  shall  touch  the  middle  of  the  bar  ;  then  pressing 
it  closely  to  the  bar,  we  move  it  in  the  direction  JVS  to  the  extremity 
of  the  bar ;  this  being  done,  we  carry  the  magnet  back  to  its  posi- 
tion, and  repeat  the  operation  several  times.     We  next  apply  the 
magnet  to  the  other  half  of  the  bar,  so  that  the  south  pole  shall  touch 
the  middle,  and  pass  it  just  as  many  times  along  this  half,  in  the 


Sphere  of  Magnetic  Activity.  215 

direction  SUV.  The  iron  thus  acquires  a  considerable  magnetic 
power.* 

This  property  is  communicated  rapidly  to  soft  iron,  but  not  dura- 
bly. Tempered  iron  receives  it  less  rapidly ;  but  retains  it  for  a 
much  longer  time. 

When  we  communicate  magnetism  with  a  single  magnet  we  call 
this  operation  the  method  of  single  touch.  For  the  method  of  dou- 
ble touch,  the  reader  must  consult  more  extensive  works.  [See 
Cambridge  Nat.  Phil.  vol.  ii.] 

458.  It  is  a  general  law  for  all  modes  of  communication,  that  the 
points  which  are  touched  last  by  one  of  the  extremities  of  the  magnet, 
take  poles  of  the  contrary  name.     Thus,  in  the  above  operation,  n 
becomes  a  north  pole,  and  s  a  south  pole. 

459.  The  magnet  loses  little  if  any  of  its  force  by  this  process, 
when  it  is  performed  f?om  point  to  point  as  we  have  described,  with- 
out changing  the  direction  on  the  same  half  of  the  bar.     We  can, 
therefore,  communicate  the  magnetic  power  to  any  number  of  iron 
bars,  with  the  same  magnet ;  and  by  uniting  these,  we  form  a  very 
powerful  magnet. 


Distribution  of  Magnetism,  and  Sphere  of  Magnetic  Activity. 

460.  As  long  as  a  piece  of  soft  iron  touches  a  magnet,  or  is  very 
near  it,  it  is  .itself  magnetic.  But  the  moment  we  remove  it,  the 
magnetism  it  had  acquired  almost  entirely  disappears. 

In  this  case  we  say  that  the  iron  is  not  magnetized  by  communi- 
cation, but  by  participation  ;  and  the  space  within  which  this  effect 
takes  place,  is  called  the  sphere  of  magnetic  activity.  Here  we  dis- 
cover an  analogy  between  magnetism  and  electricity. 

By  means  of  a  magnetized  needle  we  can  verify  the  following 
law,  which  holds  true  in  all  cases  when  iron  is  magnetized  by  parti- 
cipation. 

The  iron  acquires,  in  the  part  near  the  magnet,  a  pole  of  an  oppo- 
site name,  and  capable  of  being  attracted  by  the  nearest  pole  of  the 
magnet ;  and  consequently  the  other  extremity  of  the  iron  acquires  a 
pole  of  the  same  name  with  this  pole  of  the  magnet,  and  capable  of 
being  repelled  by  it. 

*  It  is  better  to  incline  the  magnet  to  the  bar  10°  or  12°,  than  to 
apply  it  entirely  along  the  surface. 


216  Magnetism.        ^ 

461.  This  resemblance  to  electricity  has  given  rise  to  hypotheses 
respecting  the  cause  of  magnetism,  analogous  to  those  of  Franklin 
and  Symmer  in  regard  to  electricity.* 

462.  The  above  law  has  led  to  the  invention  of  what  is  called  the 
armature  of  a  natural  magnet.     It  is  constructed  in  the  following 
manner.     Let  A  s  n  (Jig.  57)  be  a  natural  magnet,  of  which  5  is  the 
south  pole  and  n  the  north  pole.     A  piece  of  soft  iron  BC  is  shaped 
so  as  to  apply  itself  exactly  to  the  surface,  and  touch  the  two  poles. 
Just  below  these  two  poles,  the  iron  must  have  two  prominences  S, 
JV.     We  cover  the  rest  of  the  magnet  with  an  envelope  of  copper, 
DEF,  and  attach  a  ring  at  F,  by  which  to  suspend  it.     We  then 
apply  a  piece  of  soft  iron  to  the  two  prominences  /S,  JV,  in  order  to 
load  the  apparatus  more  conveniently. 

By  means  of  this  arrangement,  the  soft  iron  becomes  itself  a  mag- 
net, of  which  the  south  pole  is  S,  and  the  north  pole  JV. 

Experience  has  shown  that  an  armed  magnet  has  a  much  more 
active  and  durable  force,  than  a  common  magnet. 

*  On  the  supposition  of  two  magnetic  fluids,  M.  Coulomb  observed 
the  motions  with  all  the  precision  necessary  for  a  complete  hypothe- 
sis. He  caused  a  small  magnetized  needle  to  oscillate  at  different 
distances  from  one  of  the  poles  of  a  very  long  bar,  also  magnetized, 
but  in  a  much  less  considerable  degree.  The  effect  of  the  magnetic 
force  in  producing  these  oscillations,  is  analogous  to  that  of  gravity 
in  the  case  of  the  pendulum,  and  the  oscillations  may  equally  serve 
to  measure  the  intensity  of  this  force.  Now  by  comparing  together 
the  rapidity  of  these  oscillations,  M.  Coulomb  remarked  that  they 
became  more  and  more  slow,  as  the  needle  is  removed  further  from 
the  centre  of  the  magnetic  force,  which  proves  that  this  force  dimin- 
ishes as  the  distance  increases ;  and  from  the  law  of  this  retardation, 
he  proved  by  the  calculus,  that  magnetic  attraction  is  always  in- 
versely proportional  to  the  square  of  the  distance,  like  electric  at- 
traction and  terrestrial  gravity.  The  celebrated  astronomer,  Tobias 
Mayer,  of  Gottingen,  had  arrived  at  the  same  results. 


Declination  Needle.  217 


CHAPTER  XXXVIII. 

More  Particular  Examination  of  the  Phenomena  of  the  Magnetized 
Needle. 

463.  IF  an  unmagnetized  steel  needle  be  balanced  on  a  sharp 
pivot,  so  as  to  be  perfectly  horizontal,  the  equilibrium  will  be  de- 
stroyed upon  being  magnetized  ;  and,  what  is  very  remarkable,  its 
inclination  to  the  horizon  will  be  different  in  different  parts  of  the 
earth,  in  the  northern  hemisphere  the  north  end  of  the  needle  is 
depressed,  in  the  southern  hemisphere  the  south  end.  From  this 
phenomenon  we  must  infer  that  the  force  which  influences  the 
direction  of  the  magnetized  needle,  is  not  exerted  horizontally,  but 
in  a  direction  considerably  inclined  to  the  horizon.  If  we  would 
observe  accurately  the  phenomena  of  the  magnetized  needle,  two 
kinds  of  needles  are  necessary  ;  one  for  the  purpose  of  finding  its 
direction  in  a  horizontal  plane,  and  this  is  called  a  declination  needle  ; 
the  other  for  ascertaining  the  angle  it  makes  with  the  horizon  or 
inclination,  and  this  is  called  a  dipping  needle. 


Declination  Needle. 

464.  In  a  declination  needle  the  downward  tendency  of  the  north- 
ern extremity  is  counteracted  by  a  weight  on  the  opposite  extremity 
for  the  purpose  of  preserving  it  in  a  horizontal  position.  After  it  is 
magnetized  we  inclose  it  in  a  copper  box,  and  attach  to  it  a  graduat- 
ed paper  circle.  For  purposes  of  navigation,  this  circle  is  divided 
into  32  parts  called  points.  A  needle  constructed  in  this  way  takes 
different  names  according  to  the  purpose  to  which  it  is  destined.  If 
it  is  to  be  employed  in  determining  with  accuracy  the  direction  of 
the  needle,  we  call  it  a  declination  compass.  Its  length  varies  from 
6  to  12  inches.  The  needle  and  the  division  of  the  circle  must  be 
made  with  great  care.  If  it  is  to  be  used  in  navigation,  it  takes  the 
name  of  mariner's  compass.  With  some  slight  modifications  it  is  also 
employed  by  surveyors  in  measuring  angles  and  in  running  lines.* 

*  The  best  way  of  fitting  a  magnetic  needle  for  exact  experiments, 
is  to  suspend  it  by  a  silk  fibre,  as  it  comes  from  the  silkworm,  or  an 
assemblage  of  these  fibres  united  longitudinally.  This  kind  of  sus- 

Elem.  28 


2 18  Magnetism. 

465.  If  we  compare  the  direction  of  a  declination  needle  with  an 
astronomical  meridian,  carefully  determined,   we  shall  find  that  at 
Paris,  for  example,  the  needle  does  not  point  exactly  north,  but  de- 
viates aboiU  22J-°  towards  the  west.     For  this  reason  we  call  the 
direction  of  the  needle  the  magnetic  meridian  to  distinguish  it  from 
the  astronomical  meridian. 

466.  But  the  declination  is  different  in  different  places ;  when  we 
go  to  the  west  or  east  of  Europe,  we  observe   the  declination  to 
diminish  as  we  depart,  remaining  always  west,  however.     On  the 
continent  of  America,  we  find   a  line   running   nearly  south-east 
through  the  gulf  of  Mexico  and  Brazil,  to  the   Atlantic  ocean,  in 
which  the  declination  is  nothing.    Another  similar  line  traverses  Asia 
and  all  the  Southern  Ocean,  in  the  same  direction.     Beyond  these 
two  lines  the  needle  deviates  towards  the  east. 

467.  The  declination,  moreover,  is  not  constant  in  the  same  place. 
In  the  17th  century,  the  line  of  no  declination,  which  is  now  found 
in  America,  traversed  Europe.     Since  this  time  it  has  been  contin- 
ually moving  westward,  and  the  same  is  true  of  all  the  lines  which 
have  the  same  declination. 

It  is  obvious  that  there  must  result  from  this  motion,  changes  of 
declination  for  every  place.  It  appears  that  the  vvestern  declination 
increases  slowly  to  a  certain  limit,  which,  at  Paris,  is  about  22°  or 
23°,  after  which  it  begins  to  decrease  slowly  till  it  becomes  0, 
and  after  remaining  a  short  time  in  this  state,  the  declination 
becomes  east,  increases  to  a  certain  limit,  retrogrades,  becomes 
0,  and  so  on.  But  continued  observations  for  several  centuries  will 
be  necessary  to  determine  with  certainty  the  periods  and  laws  of  this 
motion.  From  the  observations  hitherto  made,  which  were  not  ac- 
curate till  within  a  century  and  a  half,  this  does  not  appear  to  be  very 
regular.* 

468.  Independently  of  these  great  variations,  we  observe  also  a 
slight  diurnal  motion ;  but  it  can  be  perceived  only   by  means  of 

pension  producing  no  friction  and  no  sensible  force  of  torsion,  allows 
the  needle  perfect  freedom.  It  is  similar  to  that  used  for  the  electric 
balance,  and  was  originally  invented  by  Dr  Gilbert,  an  English  phy- 


sician. 


*  It  is  not  certain  that  the  motion  is  oscillatory,  as  the  author  here 
supposes.  For  it  is  a  long  time  since  the  needle  reached  its  western 
limit,  and  it  has  uot  yet  sensibly  retrograded. 


Dipping  Needle.  219 


large  and  accurate  needles.  During  the  forenoon  the  north  end  of 
the  needle  declines  a  little  towards  the  west ;  in  the  afternoon  it 
returns  with  a  slow  motion  towards  the  east.  Graham  first  observed 
this  motion  in  1722.  Wargertin  and  Canton  repeated  and  extend- 
ed the  observations.  The  latter  proved  also  by  experiments  that 
heat  has  an  influence  upon  magnetism  which  deserves  to  be  exam- 
ined. 


Dipping  Needle. 

469.  A  dipping  needle  consists  of  a  steel  plate  several  inches 
long,  made  thin  at  the  two  extremities,  provided  with  a  short  and 
slender  axis  passing  through  the  centre  of  gravity  perpendicularly  to 
the  plate,  the  whole  being  fitted  in  such  a  manner  as  to  give  the 
needle  a  free  vertical  motion  on  its  axis.     Since  it  moves  about  its 
centre  of  gravity,  it  must,  before  being  magnetized,  remain  balanced 
in  any  position  whatever  in  which  it  is  placed.     But  when  magnet- 
ized, the  north  pole  will  he  considerably  depressed,  at  least  in  Eu- 
rope.    Its  inclination  to  the  horizon   is  measured  by  means  of  a 
graduated  circle  attached  to  the  support.     The  construction  of  such 
an  instrument  is  attended  with  many  difficulties,  and  a  good  dipping 
needle  is  rarely  to  be  found. 

470.  Much  care  also  is  required  in  the  use  of  this  instrument, 
since  it  must  be  placed  exactly  in  the  direction  of  the  magnetic  me- 
ridian, in  order  to  indicate  the  exact  inclination.     In  all  other  posi- 
tions the  inclination  will  be  too  great,  an,d  it  may  even  amount  to  90°. 
This  will  be  manifest,  if  we  direct  the  needle  in  such  a  manner  as  to 
make  a  considerable  angle  with  the  magnetic  meridian,  and  suppose 
a  thread  attached  to  its  point,  drawing  it  toward  the  magnetic  direc- 
tion.* 


*  Observations  may  be  made  in  any  direction,  without  knowing  the 
magnetic  meridian,  by  means  of  the  following  property,  which  I  be- 
lieve I  was  the  first  to  recognize. 

If  we  observe  the  inclination  of  the  needle,  reckoned  on  the  verti- 
cal, in  any  two  vertical  planes,  perpendicular  to  each  other,  the  square 
of  the  tangent  of  the  inclination  in  the  magnetic  meridian,  is  equal  to 
the  sum  of  the  squares  of  the  tangents  of  the  observed  inclinations. 

Let  H  be  the   horizontal  force  which  draws  the  needle  in   the 


220  Magnetism. 

471.  In  consequence  of  the  difficulties  attending  the  construction 
and  use  of  this  instrument,  few  accurate  observations  have    been 
made  upon  the  dip  of  the  needle.    In  Prussia  it  is  estimated  at  about 
71°.    The  result  of  the  observations  hitherto  made,  may  be  summed 
up  as  follows. 

472.  The  dip  is  subject  to  greater  variation  in  different  places 
than  the  declination.    Towards  the  north  it  increases ;  and  probably 
ther.:  is  a  place  in  North  America  from  14°  to  17°  from  the  north 
pole,  where  the  needle  is  entirely  vertical.     Towards  the  equator, 
the  inclination  diminishes ;  and  in  the  torrid  zone,  there  is  a  line  en- 
compassing the  earth,  where  the  needle  is   horizontal.     This  line 
passes  above  the  equator  in  our  hemisphere,  and  below  it  in  the 
other.     Beyond  this  line,  the  south   pole  of  the  needle  begins  to 
dip,  and  the  dip  increases  as  we  go  south  ;  probably  in  New  Zea- 
land, about  35°  or  40°  from  the  south  pole,   there  is  a  place  where 
the  needle  again  becomes  vertical.* 

magnetic  meridian,  V  the  vertical  force,  and  i  the  inclination,  reck- 
oned from  the  vertical ;  we  shall  have,  in  the  magnetic  meridian 

tang  i  —  ^. 

For  another  vertical  plane,  making  an  angle  a  with  the  magnetic 
meridian,  the  horizontal  force  will  not  be  =  //;  but  it  will  be  equal 
to  //decomposed  in  the  direction  of  this  plane;  that  is,  •=.  H  cos  a. 
The  vertical  force  will  still  be  V;  and  the  inclination  in  this  plane, 
reckoned  from  the  vertical,  being  represented  by  i'}  we  shall  have 

Hcos  a 

tang  t'  =     ~Y-  ; 

IT 

or,  substituting  for  -^  its  value  ;  tang  i'  =  tang  t  cos  a.   If  we  call  i" 

the  inclination  in  the  vertical  plane  perpendicular  to  the  preceding, 
shall  have  tang  i"  —  tang  t  sin  a  ;  squaring  these  two  equations,  and 
adding  them  together,  we  shall  have  tang  2  i'  -f-  tang  3  i"  =  tang  2  i ; 
which  is  the  property  enunciated  above. 

*  In  a  memoir  written  by  M.  Huraboldt  and  myself,  and  founded 
principally  upon  his  observations,  a  law  is  made  known  which  con- 
nects all  the  results  of  inclination  in  all  parts  of  the  earth,  and  which, 
with  a  modification  indicated  by  observation,  would  also  represent 
the  declination,  and  the  variations  of  intensity,  of  the  magnetic  forces 
in  the  different  regions  of  the  globe.  I  have  since  seen,  in  a  man- 
tiscript  of  Tobias  Mayer,  that  he  arrived  at  similar  results. 


Terrestrial  Magnetism.  221 

473.  The  dip  is  also  different  at  different  times ;  but  the  laws  of 
these  variations  are  only  deduced  from  hypothesis,  and  not  from 
exact  observations. 

Terrestrial  Magnetism. 

474.  The  phenomena  exhibited  by  the  two  needles  authorize  or 
rather  oblige  us  to  consider  the  earth  itself  as  a  great  magnet ;  since 
it  acts  upon  the  magnetized  needle  according  to  the  same  laws, 
by  which  one  magnet  acts  upon  another.     Euler  proved  that  by 
giving  to  the  magnetic  poles  of  the  earth,    the  position  which  is 
indicated  in  article  472,  the  phenomena  of  declination  and  inclina- 
tion may  be  completely  explained.     It  will  be  obvious  from  what 
has  been  said  respecting  the  reciprocal  relations  of  two  needles,  fhat 
the  magnetic  pole  situated  in  North  America,  must  take  the  name  of 
south  pole,  and  that  which  is  situated  in  New  Zealand,  the  name  of 
north  pole. 

475.  It  is  not  improbable  that  there  exists  in  the  interior  of  the 
earth  a  great  mass  of  magnetic  iron ;  for  this  metal  is  found  to  be 
diffused  in  such  abundance,  that  every  portion  of  earth  contains  more 
or  less  of  it.*     Moreover,  observations  made  with  the  pendulum, 
render  it  probable  that  the  interior  nucleus  of  the  earth,  consists 
rather  of  a  mass  of  metal,  than  of  a  merely  earthy   matter.     If  in 
addition  to  this,  we  admit  the  observations  of  Canton,  with  respect 
to  the  influence  of  heat  upon  magnetism,  we  are  led  to  believe  that 
the  diurnal  motion  of  the  sun  from  east  to  west,  may  cause  a  wes- 
terly retrogradation  of  the  magnetic  poles,  by  reason  of  which  the 
variations  of  declination  and  dip  may  be  explained  more  naturally, 
than  by  supposing  with  some  philosophers  a  particular  motion  of  the 
magnet  in  the  earth. 

*  It  is  more  simple  to  consider,  as  M.  Humboldt  and  nryself.have 
done,  the  magnetic  action  of  the  entire  earth,  as  the  resultant  of  the 
action  of  all  the  magnetic  particles,  disseminated  through  it.  Still 
all  the  hypotheses  proposed  upon  the  subject,  are  to  be  regarded 
merely  as  more  or  less  convenient  modes  of  representing  the  facts 
and  connecting  them  together. 


222  Magnetism. 


Excitation  of  Natural  Magnetism. 

476.  To  complete  the  view  which  we  have  given  of  magnetism, 
it  is  necessary  to  add  that  iron  exposed  to  the  air  for  some  time,  be- 
comes magnetized,  especially  if  it  is  placed  in  the  direction  of  the 
magnetic  meridian.     This  observation  has  given  rise  to  the  followinjr 
experiment.     We  place  a  bar  of  iron  in  the  exact  magnetic  direc- 
tion of  declination  and  dip,  and  in  a  very  little  time  the  bar  becomes 
magnetized.     The  effect  may  be  accelerated  by  rubbing  or  striking. 

477.  The  name  of  animal  magnetism  has  been  improperly  applied 
to  certain  singular  phenomena  which  take  place  in  the  human  body, 
but  which  have  no  connexion  whatever  with  the  subject  we  are  con- 
sidering. 


Addition. 

478.  I  shall  here  add  some  remarks  upon  the  analogies  which 
may  now  be  perceived  between  electricity  and  magnetism,  and  upon 
the  greater  or  less  probabilit}r,  that  these  phenomena  are  really  pro- 
duced by  the  reciprocal  attraction  and  repulsion  of  imponderable 
fluids. 

A  glance  at  the  most  simple  magnetic  phenomena,  as  the  recipro- 
cal action  of  magnets,  their  influence  upon  iron,  the  communication 
of  their  properties  by  contact,  and  even  at  a  distance,  the  develope- 
ment  of  new  poles  which  are  formed  instantaneously  in  the  points 
where  we  break  them  ;  all  this  leads  at  once  to  the  supposition  of 
two  invisible  and  imponderable  principles,  residing  naturally  in  each 
infinitely  small  particle  of  iron,  or  other  magnetic  metal,  without  the 
power,  in  any  case,  of  leaving  these  particles  to  enter  others.  In 
conformity  to  this  supposition,  when  the  magnetic  metals  are  first 
strongly  heated,  and  then  suffered  to  cool  slowly,  without  any  action 
being  exerted  upon  them,  the  two  magnetic  principles  return,  in 
each  particle,  to  the  state  of  neutrality  by  which  they  are  disguised. 
But  if  an  action  be  exerted  upon  them,  in  this  state  of  indifference, 
by  the  influence  of  another  magnetized  body,  we  observe  that  they 
are  separated  by  this  influence  in  each  particle,  one  of  them  being 
attracted  and  the  other  repelled.  We  next  find,  that  in  this  experi- 
ment, the  repulsion  takes  place  between  the  magnetic  principles  of 


Theoretical  Considerations.  223 

the  same  name,  and  the  attraction  between  principles  of  different 
names ;  and  that  both  of  these  tendencies  vary  inversely  as  the 
square  of  the  distance  ;  so  that  they  are  not  sensible  in  the  natural 
state  of  combination  of  the  two  principles,  because  these  principles 
act  with  equal  and  contrary  forces  at  equal  distances.  Guided  by 
these  laws,  we  are  able  to  measure  the  comparative  quantities  of 
free  magnetism,  existing  in  each  point  of  a  magnetized  body ;  by 
compounding  these  elementary  forces,  we  are  able  to  calculate  the 
direction  and  intensity  of  their  total  resultant ;  and  the  effort  of  this 
resultant,  although  exerted  between  single  magnetic  elements,  being 
transmitted  to  the  material  particles  of  magnetized  bodies,  by  vir- 
tue of  the  impermeability  which  retains  the  two  principles  in  each 
of  them,  indicates  the  cause,  the  law,  and  the  measure  of  the  mo- 
tions which  are  produced  in  them,  when  they  are  presented  to  each 
other  after  being  magnetized.  In  order  to  establish  the  preceding 
propositions,  we  need  not  take  any  thing  for  granted  respecting 
the  physical  nature  of  the  two  magnetic  principles.  But  if  it  be 
asked  what  this  nature  is,  it  may  be  answered,  that  there  exists  the 
most  complete  'and  perfect  analogy  between  the  laws  of  the  two  mag- 
netic principles  and  those  of  the  two  electric  principles,  so  that  the 
state  and  reciprocal  action  of  magnetized  bodies  are  exactly  similar, 
so  far  as  it  respects  the  distribution  and  law  of  the  forces,  to  those  pre- 
sented by  non  conducting  bodies,  electrified  by  influence,  and  by  the 
decomposition  of  their  natural  electricities.  Now,  when  we  exam- 
ine the  effects  of  the  electric  principles  in  the  state  of  separation  and 
freedom  in  which  we  can  obtain  them,  we  find  by  calculation  that 
their  distribution  in  conducting  bodies,  whether  free,  or  influenced 
one  by  the  other,  is  rigorously  conformable  to  the  laws  of  hydrostatic 
equilibrium,  which  two  material  fluids  would  obey,  if  they  had  no 
sensibly  gravity,  and  if  their  particles  were  endued  with  the  double 
property  of  repelling  those  of  the  same  kind,  and  attracting  those  of 
the  opposite  kind,  with  an  energy  reciprocally  proportional  to  the 
square  of  the  distance.  If  then  we  attempt  to  deduce  the  mathe- 
matical consequences  of  this  constitution,  for  those  cases  in  which  it 
is  possible  in  the  present  state  of  analytical  science,  we  obtain,  not 
vaguely,  but  rigorously  and  in  abstract  numbers,  all  the  singular  and 
minute  details  which  are  observed  when  electrified  bodies  are  pre- 
sented to  each  other,  when  they  are  removed,  and  even  when  they 
are  brought  so  near  as  to  cause  an  explosion.  Finally,  in  this  last 
case,  the  great  rapidity  with  which  the  equilibrium  is  restored  in  all 


224  Magnetism. 

parts  of  the  bodies  subjected  to  experiment,  every  point  taking  instan- 
taneously the  new  quantity  of  each  principle  required  by  the  new  state 
of  the  system  and  by  the  hydrostatic  formulas  which  express  it ;  this 
rapidity  is  itself  a  new  indication,  by  which  the  fluid  nature  of  the 
electric  principles  is  most  strongly  characterized.  It  seems,  there- 
fore, to  me,  that  in  the  present  state  of  the  theory  of  electricity,  the 
whole  honour  of  which  is  due  to  the  beautiful  analysis  of  M  Poisson, 
this  theory  itself,  by  the  fidelity  with  which  all  the  phenomena  con- 
form to  it,  furnishes  the  strongest  probability,  that  the  electric  princi- 
ples are  really  fluids,  constituted  as  this  theory  supposes.  And  ac- 
cordingly the  exact  resemblance  which  we  observe  between  the 
effects  of  the  electric  and  magnetic  principles,  in  the  cases  where  the 
first  are  subjected  to  a  coercive  force,  indicates  with  equal  probability 
that  the  two  magnetic  principles  have  also  a  similar  constitution  ; 
although  the  independence  of  the  two  classes  of  actions  does  not 
allow  us  to  suppose  them  to  be  of  the  same  nature.  Here,  then,  we 
have  proceeded  as  far  in  the  study  of  nature  as  we  are  permitted 
to  go,  since,  by  observing  the  phenomena  we  have  ascended  to  their 
experimental  laws  ;  and  from  the  laws  to  the  forces  by  which  they 
are  produced.  What  yet  remains  to  be  done  for  this  branch  of  sci- 
ence, depends  upon  the  future  perfection  of  mathematical  analysis, 
and  the  application  of  chemistry  to  the  determination  of  the  coercive 
forces,  by  which  the  magnetic  principles  are  retained  in  the  particles 
of  bodies ;  or  at  least  to  the  determination  of  the  degree  of  these 
forces  most  favourable  to  magnetic  energy. 


SECTION  VIII. 


OPTICS. 


CHAPTER  XXXIX. 

Of  Light  in  General ;  particularly  the  Phenomena  ivhich  depend 
upon  its  'Motion  in  a  Right  Line  ;  or  First  Principles  of  Optics. 

479.  THE  sense  of  touch  makes  known  different  properties  of 
bodies  in  the  most  certain  manner  ;  but  the  sense  of  sight  extends  to 
a  far  greater  number  of  objects.     We  should  have  very  few  ideas  if 
the  powers  of  our  minds  were  restricted  to  what  our  hands  are  capa- 
ble of  reaching.     The  sense  of  sight  raises  our  faculties  above  the 
limits  of  the  spot  to  which  our  bodies  are  confined,  and  introduces  us 
into  the  immensity  of  creation.     It  is  thus  one  of  the  greatest  tri- 
umphs of  the  human  mind,  that  we  have  been  able  to  extend  the 
power  of  this  sense  far  beyond  the  limits  which  nature  seems  to  have 
marked  out  for  it.     And  since  the  eye  furnishes  the  means  of  know- 
ing almost  every  thing  in  nature,  this  consideration  ought  to  give  a 
strong  interest  to  the  investigation  of  the  laws  which  govern  the  phe- 
nomena of  vision. 

480.  The  science  of  optics,  considered  as  comprehending  the 
whole  theory  of  light,  is  in  a  more  advanced  state  than  any  other 
branch  of  physics.     Its  history  is  very  important  to  the  philosopher  ; 
for  it  clearly  points  out  the  course  to  be  pursued  in  bringing  a  sci- 
ence to  its  highest  state  of  improvement.     All  the  hypotheses  that 
have  been  proposed  respecting  the  nature  of  light,  although  proceed- 
ing from  such  profound  men  as  Descartes,  Newton,  and  Euler,  have 
been  of  no  assistance  in  the  advancement  of  the  science ;  but  the 
experiments  of  Newton,  Dollond,  and  some  others,  have  led  to  a 
full  explanation  of  the  phenomena  of  Optics. 

Elem.  29 


226  Optics. 

431.  There  must  be  between  the  eye  and  the  object  seen,  some 
material  communication  by  which  a  distant  object  is  capable  of  ex- 
erting an  action  upon  the  sight.  We  do  not  know  what  this  medium 
is,  and  we  seem  destined  to  remain  ignorant,  since  we  cannot  perceive 
the  medium  itself,  but  only  the  objects  which  become  visible  by  its 
influence.  Meantime  it  is  of  no  consequence  to  us,  whether,  as 
some  of  the  ancients  thought,  this  medium  proceeds  from  the  eye  to 
the  object ;  or,  whether,  according  to  the  opinion  of  Newton,  it  pro- 
ceeds from  the  object  to  the  eye  ;  or,  whether,  according  to  the 
opinion  of  Descartes  and  Euler,  there  is  an  exceedingly  subtile  fluid, 
the  motions  of  which  produce  the  phenomena  of  vision,  in  the  same 
manner  as  the  vibrations  of  the  air  produce  the  phenomena  of  sound. 
It  is  of  little  importance  whether  any  of  these  hypotheses  be  true, 
provided  we  know  the  laws  of  the  phenomena ;  and  these  laws  have 
actually  been  developed  almost  as  perfectly  as  those  of  gravity. 

482.  The  unknown  cause  of  vision  we  call  light.  We  can  pre- 
vent the  effects  of  this  medium,  but  not  the  cause.  Light  is  pro- 
duced in  an  infinite  variety  of  ways ;  for  example,  under  all  circum- 
stances, when  steel  and  flint  are  brought  into  collision.  Even  under 
water,  steel  gives  sparks.  Electric  light  is  visible  in  water ;  and 
steel  enveloped  in  oxygen  continues  to  appear  red  under  water. 
Light  must,  therefore,  be  a  substance,  which  cannot  be  prevented 
from  penetrating  into  all  bodies,  and  which  is  capable  of  being  trav- 
ersed by  all.  It  must  be  of  a  nature  entirely  different  from  percept- 
ible substances,  since,  in  treating  of  it,  we  have  occasion  for  an  entirely 
different  system  of  mechanics  and  statics.  Indeed,  we  cannot  em- 
ploy in  optics  any  of  the  laws  of  motion  developed  in  the  preceding 
sections*.  We  shall  nowhere  find  any  indication  of  impenetrability, 

*  Here  the  author  appears  to  me  to  go  too  far.  If  we  wish  to  con- 
sider light  in  its  effects  only,  without  any  thing  hypothetical  respect- 
ing its  nature,  we  cannot  propose  to  subject  its  motions  to  mathe- 
matical calculation.  But  if  we  consider  it  as  formed  of  material 
particles,  endowed  with  a  very  rapid  rectilinear  motion,  and  capable 
of  being  attracted  by  bodies,  then  these  motions  are  subject  to  the 
laws  of  ordinary  mechanics ;  it  is  thus  that  Newton  has  deduced  by 
calculation,  the  laws  of  refraction.  Nevertheless,  this  constitution  of 
light  must  be  regarded  only  as  an  hypothesis,  to  which,  hitherto,  we 
have  been  able  to  reduce  most  of  the  phenomena  of  light ;  for,  in 
reality,  there  is  nothing  by  which  we  can  be  assured  that  light  is 


Mechanical  Phenomena  of  Direct  Light.  227 

gravity,  impulse,  &c.  If  we  may  hope  for  any  more  precise  know- 
ledge of  the  nature  of  this  substance,  we  are  to  expect  it  from  chemis- 
try ;  for  light  undoubtedly  possesses  very  remarkable  chemical  pro- 
perties. Almost  every  where  it  is  found  connected  with  heat,  which 
is  the  most  important  chemical  agent  in  nature.  Its  effects  not  only 
manifest  themselves  in  the  varied  phenomena  of  combustion,  but  in 
most  of  the  electrico-chemical  experiments.  The  chemist  also 
observes  in  the  natural  properties  of  certain  substances,  several 
changes  which  can  only  be  produced  by  the  action  of  light.  Finally, 
no  one  can  fail  to  perceive  the  great  and  beneficent  influence  of  light 
upon  organized  bodies.  But  the  laws  of  its  chemical  action  are  as 
obscure,  as  the  laws  of  its  motion  are  simple. 


Mechanical  Phenomena  of  Direct  Light. 

483.  The  sun,  flame,  and  bodies  in  a  state  of  combustion,  emit 
light  in  all  directions.     Such  bodies  are  said  to  be  self-luminous. 
Other  bodies  simply  send  back  the  light  which  they  receive.    These 
are  said  to  be  illuminated.     Light  penetrates  through  all  the  gases, 
most  liquids,  particularly  water,  and  many  solid  bodies,  among  which 
glass  is  particularly  distinguished.     Such  bodies  are  called  transpa- 
rent ;  others  retain  the  light,  and  are  called  opaque. 

484.  The  first  law  relating  to  the  motion  of  light,  is  the  following ; 
In  a  transparent  homogeneous  medium  the  transmission  of  light 

takes  place  in  straight  lines. 

There  is  no  need  of  any  particular  experiment  to  demonstrate  this 
law.  Its  truth  is  evident  from  the  following  observation,  which  may 
be  repeated  at  pleasure.  It  is  impossible  to  see  an  object  if  an 
opaque  body  is  interposed  in  the  straight  line  drawn  from  this  object 
to  the  eye  ;  also  where  a  room  is  shut  up  so  as  to  admit  no  light, 
except  through  a  small  aperture,  the  illuminated  particles  of  dust 
appear  to  be  arranged  in  straight  lines. 

485.  By  this  law  the  effects  of  direct  light  are  perfectly  repre- 
sented, and  the  science  of  optics  is  reduced  within  the  province  of 


composed  of  material  particles  sent  forth  from  the  luminous  body, 
and  many  analogies  tend  to  represent  it  as  merely  the  effect  of  vibra- 
tions transmitted  in  the  manner  of  sound,  through  an  elastic  medium. 


228  Optics. 

geometry.     A  straight  line,  considered  as  the  path  described  by 
light,  is  called  a  ray. 

486.  From  each  point  of  a  luminous  body  the  rays  proceed  in 
every  direction  in  which  straight  lines  can  be  drawn  in  a  transparent 
medium ;  and  each  ray  of  light  passes  on  in  a  straight  line,  until  it 
encounters  a  medium  of  different  material  properties  ;  then  the  direc- 
tion changes  according  to  the  nature  of  the  body  encountered. 

487.  If  the  ray  enters  a  transparent  medium  more  rare  or  more 
dense,  or  of  which  the  material  properties  are  different,  it  undergoes 
a  refraction  ;  that  is,  it  is  more  or  less  deflected  from  its  rectilinear 
direction.     The  laws  of  these  phenomena  constitute  what  is  called 
dioptrics. 

If  the  ray  encounters  the  polished  surface  .of  an  opaque  body,  it  is 
reflected  in  a  determinate  direction.  The  laws  of  this  reflection  are 
comprehended  under  the  part  of  optics  called  catoptrics.  If  a  ray 
passes  very  near  a  body,  it  undergoes  a  feeble  inflection,  the  laws  of 
which  are  not  perfectly  understood,  but  this  phenomena  does  not 
appear  to  have  any  very  important  influence  on  the  phenomena  of 
vision.  For  this  reason  it  will  be  sufficient  to  have  merely  mention- 
ed the  fact  in  this  place. 

Lastly,  if  light  falls  upon  an  opaque,  unpolished  body,  it  under- 
goes changes  which  we  must  here  examine  with  attention. 

488.  In  the  case  last  supposed,  the  body  is  illuminated,  that  is, 
all  its  points  become  luminous,  because  it  reflects  the  light  which  it 
'receives,  towards  all  the  points  to  which  a  straight  line  can  be  drawn 
through  the  transparent  medium.* 

489.  It  is  evident  that  a  considerable  diminution  must  always  re- 
sult from  this  dispersion  of  light,  since  each  ray  is  subdivided,  as  it 
were,  into  an  infinite  number  of  rays.     Accordingly  the  impression 
made  by  this  disseminated  light  is  incomparably  less  strong  than  the 
dazzling  light  of  self-luminous  bodies. 

490.  But  independently  of  this  dissemination,  the  light  is  dimin- 
ished by  another  cause.     Remarkable  changes  almost  always  take 
place  in  light,  by  the  contact  of  bodies.     There  are  some  bodies 
which  send  off  nearly  all  the  light  they  receive.     These  appear  per- 
fectly white.     Others  reflect  very  little  or  none.     These  are  per- 
fectly black.     In  all  others  the  light  undergoes  a  particular  change, 

*  Some  particulars  relating  to  this  subject  will  be  found  in  what  is 
subjoined  at  the  end  of  the  section. 


Mechanical  Phenomena  of  Direct  LAght.  229 

which  may  be  considered  as  a  chemical  modification  of  the  luminous 
matter.  Dispersed  light  makes  an  impression  upon  the  eye  alto- 
gether different  from  that  of  primitive  light.  We  call  this  impres- 
sion colour.  The  primitive  light  is  always  diminished  when  it  is 
thus  modified.  This  effect  is  less  considerable  in  bright  and  lively 
colours,  than  in  those  which  are  obscure  and  faint.  We  shall  exam- 
ine the  phenomena  of  colour  more  fully  in  a  subsequent  chapter. 
We  may,  however,  remark  in  this  place,  that  colour  does  not  belong 
to  bodies,  but  that  it  is  the  reflected  light  itself  which  is  blue,  green, 
red,  &c. ;  since  the  sensations  of  different  colours  cannot  be  convey- 
ed to  the  eye  without  this  light. 

491.  A  well  known  experiment  proves  that  the  light  which  comes 
from  a  coloured  body,  has  itself  colour.  In  a  darkened  room,  when  the 
light  of  an  illuminated  object  passes  through  an  aperture,  and  falls 
on  a  white  wall,  the  objects  are  represented  inverted  and  indistinct, 
yet  they  appear  with  their  natural  colours.  This  phenomenon  is 
easily  explained  on  the  supposition  of  the  rectilinear  motion  and  col- 
our of  light.  Suppose  some  object,  a  large  straight  staff,  for  exam- 
ple, painted  with  different  colours,  and  placed  at  some  distance  in 
front  of  a  very  dark  room,  through  which  the  light  can  penetrate 
only  through  one  small  opening,  and  suppose  this  triangular.  Let 
the  staff  be  placed  in  such  a  manner,  that  the  light  proceeding  from 
it,  shall  enter  the  room  through  this  opening,  and  let  there  be  in  the 
chamber  a  perfectly  white,  smooth  wall,  facing  the  opening,  so  that  the 
light  which  passes  through  the  opening  may  fall  upon  this  wall.  If 
we  first  observe  the  light  which  comes  from  the  top  of  the  staff, 
which  we  suppose  red,  we  remark  that  this  light  has  the  form  of  a 
triangular  pyramid,  of  which  the  most  brilliant  point  is  at  the  vertex, 
and  the  sides  and  angles  of  which  are  determined  by  the  form  of 
the  opening.  The  light  of  this  pyramid  will  strike  the  white  wall 
towards  the  bottom,  and  will  illuminate  a  small  space  of  a  triangular 
form  like  the  opening.  This  illuminated  space  is  the  indeterminate 
image  of  the  end  of  the  staff,  and  since  this  image  is  red,  it  follows 
that  the  light  which  produces  it  must  also  be  red.  If  we  suppose 
the  lower  part  of  the  staff  to  be  blue,  it  will  produce  towards  the  top 
of  the  wall,  a  similar  triangular  blue  image,  rather  confused  and  in- 
distinct. The  same  is  true  of  all  the  points  of  the  staff.  Hence  we 
see  that  an  inverted  image  of  the  whole  staff  must  be  painted  on  the 
wall,  which  is  not  composed  of  luminous  points,  but  of  small  trian- 
gles of  light.  If  the  opening  were  quadrangular,  the  image  would 


230  Optics. 

be  formed  of  small  luminous  squares ;  if  it  were  round,  of  small 
circles,  &c.  The  image  will  be  the  more  confused,  according  as 
the  aperture  is  greater,  the  object  nearer,  and  the  wall  which  re- 
ceives the  image,  more  remote. 

When  the  sun  appears  through  a  thick  foliage,  and  we  receive  the 
image  of  its  light  upon  a  plane  perpendicular  to  the  direction  of  its 
rays,  the  illuminated  places  are  all  circular,  but  not  defined  with  pre- 
cision. These  are  also  indeterminate  images  of  the  sun  which  are 
formed  in  the  same  manner. 

492.  This  single  observation,  that  coloured  light  is  much  more 
feeble  than  white  light,  would  lead  us  to  suppose  that  the  white  light 
of  the  sun  is  a  mixture  of  different  coloured  lights ;  and  that  the  sur- 
face of  each  body  reflects  only  some  of  its  constituent  principles,  that 
is,  only  some  of  its  colours,  while  it  absorbs  others  and  renders  them 
ineffectual.     This  opinion  will  be  entirely  confirmed  by  the  theory 
of  dioptric  colours. 

It  is  a  known  fact,  that  some  bodies  absorb  white  light,  and  after- 
wards send  it  out  in  a  dark  place. 

As  to  the  direct  motions  of  coloured  light,  they  are  subject  to  the 
same  laws  as  those  of  white  light. 

493.  Kepler  supposed  that  the  transmission  of  light  was  instanta- 
neous, that  is,  that  its  velocity  was  too  great  to  be  measured.     Nev- 
ertheless, since  his  time,  astronomers  have  actually  measured  this 
velocity  ;  they  have  observed  that  the  eclipses  of  Jupiter's  satellites, 
take  place  so  much  the  later,  according  as  this  planet  is  further  from 
us.     From  this  phenomenon  they  have  calculated  that  light  passes 
through  the  diameter  of  the  earth's  orbit,  or  190000  000  miles  in 
about  1C  minutes,  and  that  so  far  as  we  can  judge,  this  motion  is 
perfectly  uniform.     Thus  in  one  second,  light  passes  through  a  space 
of  about  200  000  miles. 

494.  It  is  difficult  to  determine  whether  the  intensity  of  the  same 
luminous  ray  decreases  or  remains  constant,  when  it  passes  through 
a  vacuum  or  a  perfectly  transparent  medium.     The  great  intensity, 
however,  of  the  light  of  the  fixed  stars,  compared  with  their  immense 
distance,  renders  the  latter  opinion  the  more  probable  ;  *  it  at  least 
proves  beyond  a  doubt,  that  in  the  smaller  distances  which  light 
describes,  it  does  not  suffer  any  sensible  diminution. 


*  According  to  the  opinion  of  astronomers,  light,  notwithstanding 
its  prodigious  velocity,  employs  at  least  three  years  in  coming  from 
the  nearest  fixed  star. 


Mechanical  Phenomena  of  Direct  Light.  231 

495.  But  the  light  which  proceeds  from  a  body  loses  its  intensity 
by  diffusing  itself,  since  it  is  spread  over  a  space  the  more  extended 
the  farther  it  travels  from  the  body.     By  means  of  some  well  known 
geometrical  theorems,  it  may  be  demonstrated,  that  the  intensity  of 
light  is  inversely  proportional  to  the  square  of  the  distance  ;  on  the 
supposition  that  it  is  not  diminished  by  any  other  cause,  except 
the  divergence  of  the  rays.     This  is  the  principal  theorem  relative 
to  the  intensity  of  light.* 

496.  But  independently  of  the  distance,  the  brightness  of  the 
light  is  modified  by  several  causes,  among  which  are  the  following  ; 
1.  The  intensity  of  the  light  of  the  illuminated  body.     2.  Its  mag- 
nitude and  position.     3.  The  situation  of  the  plane  which  receives 
the  light.    4.  The  properties  of  the  medium  through  which  the  light 
passes. 

We  should  be  careful  to  distinguish  the  intensity  of  illumination 
from  the  intensity  of  the  light  itself;  for  the  first  depends,  as  we 
have  already  shown,  upon  the  quantity  which  it  absorbs. 

497.  In  the  construction  of  optical  instruments,  brightness  is  an 
object  of  particular  attention,  because  it  is  necessary  to  know,  not 

*  Let  A  (Jig.  58)  be  a  radiating  point,  and  at  the  distance  AB, 
suppose  a  geometrical  plane  BCD,  perpendicular  to  AB.  It  is  ob- 
vious that  the  light  falling  from  A  upon  BCD,  must  have  the  form 
of  a  pyramid,  whose  vertex  is  A,  and  whose  base  is  BCD.  If  we 
prolong  this  pyramid  indefinitely,  and  at  some  distance  AE,  taken 
at  pleasure,  cut  it  by  a  plane  EFG,  parallel  to  the  former  BCD,  it  is 
obvious  that  there  must  be  as  much  light  in  one  of  these  planes  as  in 
the  other.  But  BCD  is  smaller  than  EFG ;  and  consequently,  the 
light  must  be  more  concentrated  in  the  former,  in  the  ratio  of  the 
two  surfaces.  Let  L  be  the  intensity  in  BCD,  and  I  that  in  EFG. 
Then  L  :  I  : :  EFG  :  BCD.  Now  EFG  and  BCD,  being  parallel 
sections  of  a  pyramid,  are  similar  polygons  ;  consequently,  they  are 
to  each  other  as  the  squares  of  their  homologous  sides ;  that  is, 
EFG  :  BCD  :  :  EF2  :  BC2.  But  AEF  and  ABC  are  also  similar 
triangles,  since  the  lines  BC,  and  EF  are  parallel ;  we  have  then 

EF  :  BC  : :  AE  :  AB,  or  EF2  :  BC2  :  :  AE2  :  AB2  ; 

whence,  EFG  :  BCD : :  AE2 :  AB2.  Therefore,  L :  I : :  AE2 :  AB2. 
If  A  contain  several  luminous  points,  the  law  of  each  would  be  the 
same  ;  hence  the  law  demonstrated  is  true  generally  for  all  luminous 
bodies  ;  provided  we  estimate  separately  the  distance  of  each  point 
of  the  body. 


232  Optics. 

only  what  is  the  intensity  of  light  out  of  the  eye,  but  also  over  what 
space  the  light  is  distributed  in  the  eye,  by  the  refraction  of  the 
glasses. 

498.  The  total  absence  of  light  is  called  darkness.     In  an  illumi- 
nated space,  we  give  the  name  of  shadow  tp  places  which  the  light 
of  the  luminous  body  is  incapable  of  reaching  directly,  in  conse- 
quence of  the  interposition  of  some  opaque  body.  Behind  each  opaque 
body  there  is  always  a  space,  upon  which  the  light  of  the  luminous 
body  cannot  immediately  fall ;  so  that  an  eye  placed  there  would  not 
be  able  to  perceive  the  luminous  body.     We  call  this  space  perfect 
shadow.  But  if  the  luminous  body  does  not  consist  of  a  single  luminous 
point,  like  the  fixed  stars ;  but,  like  the  sun,  moon,  and  flame,  is  of  an 
apparently  sensible  magnitude,  there  will  also  be,  behind  the  opake 
illuminated  body,  a  space  which  will  receive  only  a  part  of  the  light. 
An  eye  placed  there,  would  see  a  greater  or  less  portion  of  the  lumi- 
nous body.     We  call  this  space  the  penumbra.     It  is  obvious  that 
the  gradations  between  the  perfect  shadow  and  the  space  entirely 
illuminated,  succeed  each  other  in  such  a  manner  that  in  observing 
the  form  of  the  shadow,  we  are  unable  to  perceive  a  well-defined 
limit.     Meantime,  as  the  shadow  depends  simply  on  the  form  of  the 
luminary,  on  that  of  the  body  illuminated,  and  on  the  rectilinear  mo- 
tion of  light,  this  theory  is  susceptible  of  a  rigorous  mathematical 
demonstration  ;  that  is,  we  can  demonstate  for  each  case  the  form 
of  the  perfect  shadow  and  of  the  penumbra,  as  well  as  the  intensity 
of  the  light,  at  each  point  of  this  last.    Ordinarily,  we  understand  by 
the  word  shadow,  the  configuration  of  the  shaded  space,  which,  be- 
comes visible  upon  a  second  opaque  body  placed  behind  the  first. 

499.  We  ought  not  to  close  this  chapter  without  saying  something 
of  the  relations  which  exist  between  light  and  heat.     Solar  light  and 
terrestrial  fire  exhibit  both  combined  together.     In  other  circum- 
stances, light  appears  without  heat;  or  more  frequently,  heat  \vith- 
out  light.     The  more  we  consider  their  effects,  the  more  we  are 
induced  to  regard  them  as  two  entirely  distinct  substances. 


Vision.  233 

CHAPTER  XL. 

Vision. 

500.  ALTHOUGH  a  complete  theory  of  vision  includes  not  only 
dioptrics,  but  also  a  description  of  the  eye,  we  shall  only  state  briefly 
what  takes  place  in  the  eye  in  the  case  of  vision. 

The  eye  itself  is  a  globe  provided  with  various  coatings,  and 
placed  in  a  cavity,  in  which  it  can  move  freely  in  all  directions,  by 
means  of  the  muscles  attached  to  it.  The  exterior  coating  is  com- 
posed of  a  white,  opaque,  and  horny  substance  ABCD  (Jig.  59),  and 
is  called  the  sclerotica.  But  in  the  from  part  of  this  coating,  between 
A  and  D,  where  its  curvature  is  increased,  and  where  it  is  perfectly 
transparent,  it  takes  the  name  of  cornea.  Within  the  sclerotica  we 
find  the  choroid  coat  composed  of  a  dark-coloured  substance  ;  and  un- 
der this  is  the  retina,  which  is  a  white  membrane,  thin  and  almost  vis- 
cous, and  which  is  generally  considered  as  the  seat  of  the  sensation. 
This  membrane  is  formed  by  the  continuation  of  the  medullary  part 
of  the  optic  nerve,  which  proceeds  from  the  brain,  and  enters  the 
eye  at  BC.  Behind  the  cornea  JJD,  the  choroid  is  detached  and 
divided  into  two  parts,  one  of  which  is  in  the  shape  of  a  ring,  and 
forms  the  circular  opening  called  the  pupil.  The  membrane  which 
forms  this  ring  has  received  the  name  of  iris,  on  account  of  the 
variety  of  its  colours.  It  consists  of  a  very  delicate  tissue  of  con- 
tractile fibres,  by  which  the  pupil  is  diminished  when  the  eye  is 
affected  by  strong  light,  and  which  return  to  their  primitive  state 
when  the  light  is  feeble.  These  operations  take  place  independently 
of  the  will,  and  even  without  our  being  sensible  of  them. 

Behind  the  iris  is  a  body  EF,  of  considerable  consistency,  trans- 
parent, and  shaped  like  a  lens,  which  divides  the  interior  of  the  eye 
into  two  unequal  spaces,  called  the  anterior  and  posterior  chambers 
of  the  eye.  This  body  is  called  the  crystalline.  The  anterior 
chamber  contains  a  transparent  liquid  like  water,  called  the  aqueous 
humour.  The  posterior  chamber  is  filled  with  a  transparent  and  ge- 
latinous matter,  which  is  called  the  vitreous  humour. 

A  line  GB,  which  passes  through  the  pupil  perpendicular  to  the 
two  faces  of  the  crystalline,  is  called  the  axis  of  the  eye.  In  a  well 
formed  eye,  this  axis  is  directed  to  the  object  at  which  we  look,  so 
that  the  strongest  sensation  of  vision  is  produced  at  B. 

Elem.  3G 


234  Optics. 

501.  Vision  takes  place  through  the  medium  of  a  small,  inverted, 
but  very  distinct  image  of  the  object,  on  the  retina.  Let  HI 
(fig.  59)  be  the  object  to  which  the  eye  is  directed.  Each  of  its 
points  will  send  forth  rays  in  all  directions.  Let  us  take  the  point 
G.  A  small  part  of  its  rays  penetrate  through  the  pupil  into  the 
interior  of  the  eye,  forming  a  cone  which  is  indicated  by  three  lines 
in  the  figure.  The  ray  in  the  middle  of  this  cone  traverses  the  eye 
without  deviating  from  its  direction,  and  marks  on  the  retina  the 
point  which  represents  G.  The  other  rays  which  surround  it  are 
refracted,  but  in  such  a  manner  that  they  all  unite  at  the  same 
point  g.  Now  if  G  were  blue,  for  example,  g  would  receive  only 
blue  light,  and  would  itself  be  blue.  This  would,  therefore,  be  an 
image  of  the  point  G.  The  same  takes  place  for  all  the  rays  which 
come  from  any  point  of  the  object,  from  H  or  /,  for  example. 

If  then  we  draw  from  each  luminous  point,  a  straight  line,  nearly 
through  the  middle  of  the  crystalline,  we  can  find  the  place  where 
this  point  is  represented  on  the  retina.  Thus  H  will  be  painted  in 
h,  and  I'm  i.  Hence  we  see  how  a  small  inverted  image  will  be 
painted  on  the  retina. 

502.  All  that  is  required  for  the  explanation  of  this  phenome- 
non, will  be  perfectly  deduced  from  the  principles  of  refraction. 
But  there  still  remain  some  important  questions  in  anatomy  and 
physiology  to  be  solved. 

Experience  shows  that  we  do  not  see  objects  distinctly,  which  are 
either  very  near  or  very  distant ;  and  that  for  all  eyes,  there  is  a  cer- 
tain distance,  at  which  vision  is  most  perfect.  This  is  entirely  con- 
formable to  the  laws  of  dioptrics  j  for  we  can  demonstrate  by  the 
principles  of  refraction,  that  the  rays  of  a  point  too  near  would  not 
unite  exactly  at  the  retina,  but  a  little  behind  it ;  the  rays  of  an  ob- 
ject too  distant,  on  the  contrary,  would  have  their  point  of  meeting  a 
little  in  front  of  the  retina  ;  and  in  both  cases  there  would  not  be  a 
distinct  image.  But  we  know  by  experience,  that  in  the  eye,  nature 
has  remedied  this  defect  to  a  certain  degree.  But  there  has  been 
much  dispute  as  to  the  manner  in  which  this  is  effected.  Some 
think  that  the  crystalline  is  capable  of  moving  a  little  backward  and 
forward.  But  it  is  more  probable  that  the  curvature  of  its  surface 
admits  of  slight  variations.* 

*  The  celebrated  anatomist,  Home,  supposes  that  the  four  straight 
muscles  which  move  the  eye,  change  also  the  curvature  of  the  cornea, 
which  is  very  elastic  ;  and  that,  by  this  means,  we  are  able  to  sec 


Fmon.  23S 

The  distance  of  distinct  vision  is  very  different  in  different  per- 
sons. Ordinarily  it  is  about  8  inches.  Those  persons  with  respect 
to  whom  it  is  least,  are  called  myopes,  or  short-sighted.  Those  in 
whom  it  is  greatest,  are  called  presbytes,  or  long-sighted.  But  these 
peculiarities  result  more  frequently  from  habit,  than  from  the  origi- 
nal structure  of  the  eye.* 

distinctly  objects  which  are  placed  at  different  distances.  There  are 
animals  which  must  have  the  faculty  of  distinguishing  very  distant 
objects,  without  losing  the  power  of  seeing  them  when  they  are 
brought  near.  Birds  of  prey,  for  example,  which  are  able,  from  a 
great  elevation,  to  spy  out  a  small  animal  upon  the  surface  of  the 
earth,  and  to  continue  to  see  it  till  they  are  ready  to  pounce  upon  it, 
must  necessarily  change  the  form  of  the  eye ;  and  in  fact,  if  we  ex- 
amine the  structure  of  this  organ,  we  observe  that  the  sclerotica, 
which  is  thin  in  the  back  part,  is  f  irnished  in  frout  with  a  bony  cir- 
cle, composed  of  small  pieces  which  are  capable  of  playing  upon 
each  other,  and  which  offer  a  very  firm  support  for  the  attachment 
of  muscles.  Generally,  the  organ  of  vision,  in  different  animals,  is 
necessarily  adapted  to  their  mode  of  life.  For  example,  fishes  which 
live  in  a  medium  in  which  light  passes  with  greater  difficulty  than  in 
the  air,  and  is  in  a  great  measure  absorbed,  have  eyes  that  differ  in 
their  structure  from  those  of  land  animals.  The  crystalline  is  spheri- 
cal, and  more  deeply  imbedded  in  the  vitreous  humour,  and  sometimes 
moveable.  The  cornea  is  almost  always  flat ;  and  often  the  pupil, 
instead  of  a  circular  opening,  presents  a  sort  of  lattice,  by  means  of 
notches  in  the  iris,  which  is  moveable.  We  are  ignorant  in  what 
manner  these  modifications  are  favourable  to  vision,  although  it  is 
probable  that  they  are  suited  to  the  subject.  It  may  be  observed 
also  that  the  crystalline  is  round  in  the  cormorant,  a  bird  which  dives 
for  fish. 

*  Short-sighted  persons,  for  the  most  part,  have  very  large,  promi- 
nent eyes,  in  consequence  of  the  great  convexity  of  the  cornea.  Now 
this  convexity  supposes  a  greater  interval  for  the  aqueous  humour 
contained  in  the  anterior  chamber  of  the  eye,  and  therefore  a  larger 
space  between  the  anterior  convex  part  of  the  crystalline  and  the 
posterior  concave  part  of  the  cornea.  Hence  the  object  must  be 
nearer  the  eye,  in  order  that  the  luminous  rays  may  converge  suffi- 
ciently. In  long-sighted  persons,  on  the  contrary,  the  aqueous  hu- 
mour occupies  less  space,  and  this  is  what  we  observe  in  those  who 
are  advanced  in  life.  This  defect,  thereforp,  results  less  frequently 
from  habit,  than  from  the  structure  of  the  eye,  although^U  ' 
doubtedly  increased  by  habit. 


236  Optics. 

503.  The   most  unaccountable  circumstance  in  the  phenomena 
of  vision  is,  that  the  image  which  produces  the  sensation  is  in  the 
eye,  while  the  image  which  we  see  is  without.     The  cause  of  this 
depends  undoubtedly  upon  the  power  of  the  imagination,  and  con- 
sequently belongs  to  psychology.    In  the  mean  time,  though  this  phe- 
nomenon is  not  yet  explained,  and  perhaps  never  will  be,  yet  the 
fact  is  so  certain  that  it  may  serve  as  a  fundamental  principle  in  the 
explanation  of  other  phenomena. 

504.  The  sensation  and  the  judgment  we  form  in  consequence  of 
it,  are  so  confounded  by  habit,  that  we  often  think  we  experience  a 
sensation,  when  we  only  pass  a  judgment  upon  one.     This  is  most 
frequently  the  case  with  respect  to  the  sense  of  sight.     For  this  rea- 
son it  is  necessary  to  distinguish  carefully  in  the  case  of  vision,  what 
is  a  true  sensation,  unmixed  with  any  inference.     To  do  this,  we 
must  take  the  most  simple  case,  which  is  undoubtedly  that  in  which 
a  single  radiant  point  sends  light  to  the  eye.     What  we  remark  in 
this  case,  is  the  colour  of  the  light,  and  the  direction  in  which  the 
middle  ray  of  the  luminous  cone,  strikes  the  retina.     These  are  the 
two  most  simple  elements  of  the  sensation  of  vision. 

505.  The  unquestionable  facts  stated   in  the  two  preceding  arti- 
cles, are  sufficient  to  solve  the  question   so  often  discussed,  how  it 
happens  that  we  see  objects  erect,  when  th*,  image  is  inverted  on  the 
retina.     If  the  image  of  the  point  H  which  we  see,  were  at  the  same 
place  where  the  sensation  is  produced   in  the  eye,  that  is,  at  A,  we 
should  perceive  the  whole  object  as  a  thing  situated  in  the  eye, 
and  \fle  should  certainly    see  it  in  the  position  which   the    image 
takes  in  the  eye,  that  is,  inverted  ;  but  as  we  are  sensible  of  the  col- 
our and  the  direction  of  the  light  coming  from  H,  the  visible  image 
advances  and  recedes  by  an  inexplicable  effect  of  the  force  of  our 
imagination,  so  that  we  cannot  see  the  point  Hany  where,  except  in 
the  part  of  the  line  Hh  which  is  without  the  eye  ;  that  is,  a  point 
the  representation  of  which  is  below  the  axis  of  the  eye,  is  seen 
above,  and  vice  versa. 

506.  If  we  see  objects  single,  with  two  eyes,  it  is  because  we 
always  see  objects  with  both  eyes  at  the  same  time,  and  the  two 
images  are  confounded. 

507.  The  apparent  magnitude  of  an  object  HI,  (Jig.  59)  is  pro- 
perly the  magnitude  of  its  image  h  i  on  the  retina.     If  we  bring  the 
object  nearer,  its  image  appears  larger  ;  if  we  remove  it,  its  image 
appears  smaller ;  the  apparent  magnitude  is  therefore  entirely  differ- 


\ 


Fm'ow.  237 

ent  from  the  real  magnitude,  for  this  is  invariable.  It  is  obvious 
that  the  apparent  magnitude  increases  or  diminishes  with  the  angle 
HLI;  consequently,  we  consider  the  angle  HLI,  under  which  we 
see  an  object,  as  the  measure  of  its  apparent  magnitude  ;  and  we 
call  it  the  visual  angle  or  apparent  diameter  of  the  object. 

508.  If  the  visual  angle  is  too  small,  we  are  not  in  a  state  to  dis- 
cern the  object.     We  generally  say,  that  an  object  ceases  to  be  visi- 
ble, when  its  visual  angle  is  smaller  than  one  minute.    This  estimate, 
however,  can  only  be  considered  as  approximate ;  for  the  visibility 
of  a  point  or  object,  does  not  depend  upon  the  magnitude  of  the 
visual  angle  alone,  but  also  upon  the  manner  in  which  the  light  of 
the  object  detaches  itself  from  the  light  of  the  ground  on  which  it  is 
seen.     If  a  strongly  illuminated  object  is  placed  on  a  dark  ground,  it 
may  be  visible  under  an  angle  smaller  than  a  second  ;  this  is  proved 
by  the  observation  of  the  fixed  stars,  among  which  there  is  perhaps 
no  one  which  has  an  apparent  diameter  of  one  second  ;  but  when  the 
light  of  an  object  is  less  detached  from  the  ground  upon  which  it  is 
seen,  it  may  be  invisible  under  a  much  greater  angle. 

509.  The  eye  cannot  discern  immediately  the  distance  of  objects ; 
for  the  impression  which  is  made  upon  the  retina,  depends  solely 
upon  the  direction  and  intensity  of  the  luminous  rays  at  the  instant  of 
contact.     Accordingly,  the  greater  or  less  space  described  by  the 
ray  before  reaching  the  eye,  can  have  no  influence  upon  the  sensa- 
tion produced.     Consequently,  what  is  indicated  by  both  eyes  as  to 
the  distance  of  an  object,  is  not  a  sensation  but  a  judgment,  which  is 
so  confounded  by  habit  with  the  sensation,  that  we  can  hardly  dis- 
tinguish them. 

510.  But  nature  has  greatly  facilitated  our  estimate  of  distances; 
for,  although  distance  is  not  immediately  perceived,  there  are  never- 
theless, circumstances  connected  with  the  sensation  which  differ  ac- 
cording as  the  object  is  nearer  or  more  remote.     Among  these  the 
following  may  be  mentioned. 

(1.)  In  well  formed  eyes,  the  axis  of  each  eye  must  be  directed 
towards  the  point  which  we  consider.  When  this  point  is  near,  the 
two  axes  must  make  a  much  greater  angle  than  when  it  is  distant, 
and  the  effort  of  the  muscles  which  is  necessary  to  produce  the  mo- 
tion of  the  pupil,  is  in  fact  something  to  be  felt. 

(2.)  The  degree  of  distinctness  and  precision  with  which  we  see 
each  point,  is  different  for  a  near  object  from  what  it  is  in  the  case 
of  a  more  remote  one. 


238  Optics. 

(3.)  The  light  of  a  distant  object,  other  things  being  the  same,  is 
more  feeble  than  that  of  a  near  one,  on  account  of  the  light  being 
more  diffused  ;  and  also  on  account  of  the  imperfect  transparency  of 
the  air,  especially  in  the  lower  regions  of  the  atmosphere. 

(4.)  The  apparent  magnitude  of  an  object  of  which  we  know  the 
real  magnitude,  determines  our  estimate  of  the  distance. 

(5.)  The  position  of  an  object  with  respect  to  other  objects,  the 
distance  and  situation  of  which  are  known,  serves  also  to  assist  our 
judgment. 

511.  For  objects  at  a  small  distance,  where  an  exact  estimate 
of  the  distance  is  most  important  to  us,  all  these  circumstances  are 
combined  in  forming  our  judgment.     The  greater  the  distance,  the 
less  certain  is  our  estimate,  and  beyond  the  region  of  die  atmosphere 
all  these  means  fail  entirely,  so  that  not  only  elevated  meteors,  but 
even  the  stars  appear  to  be  situated  in  the  same  surface  ;  that  is, 
attached  to  the  blue  vault  which  the  light  of  the  air  represents  to  us. 
According  to  the  observations  in  article  510,  we  are  able  to  explain 
why  this  vault  appears  to  have  the  form  of  a  segment  less  than  a 
hemisphere,  and  why  the  sun,  moon,  and  stars  appear  larger  and 
more  distant  from  each  other  when  in  or  near  the  horizon,  than  when 
they  are  in  the  more  elevated  parts  of  the  heavens.     There  are  sev- 
eral circumstances  which  conspire  to  give  us  the  impression  that  ob- 
jects seen  near  the  horizon,  other  things  being  the  same,  are  more 
distant,  many  of  these  circumstances  being  wanting,  when  the  object 
is  at  a  great  altitude. 

512.  The  two  last  circumstances  mentioned  in  article  510,  as  a 
means  of  judging  of  distance,  serve  also  to  determine  the  real  mag- 
nitude of  an  object ;  that  is,  if  we  are  able  to  judge  of  the  distance 
by  other  means,  this  distance  compared  with  the  apparent  magnitude 
of  the  object,  affords  an  indication  of  its  real  magnitude. 

Moreover,  if  we  see  an  unknown  object  amongst  others  which  are 
known,  these  furnish  the  means  of  estimating  its  real  magnitude.  . 

513.  In  like  manner  the  apparent  form  of  a  thing  is  not  its  real 
form.     The  image  which  is  painted  on  the  retina,  is  not  a  body,  but 
a  plane,  and  consequently  each  object  appears  to  our  eyes  as  a  sim- 
ple surface.     Yet  we  judge  with  great  exactness  of  the  real  form  of 
an  object,  especially  when  it  is  very  near,  since  every  tiling  which 
enables  us  to  determine  the  magnitude  and  distance  of  an  object, 
helps  us  also  to  judge  of  its  form.     But  the  best  indications  of  form 
are  the  alternations  of  light  and  shade,  especially  when  we  are  able 
to  consider  the  object  on  more  than  one  side. 


Reflection  of  Light.  239 

514.  From  these  observations  we  may  conceive  the  possibility  of 
representing  apparent  objects  upon  a  simple  surface,  as  is  done  in 
painting.     Independently  of  what  the  inventive  genius  of  the  artist 
does  for  the  picture,  it  is  necessary  that  the  rules  of  geometrical  and 
aerial  perspective  should  be  observed.     The  first  teaches  us  how  to 
draw  the  outlines  of  objects,  as  they  appear  to  the  eye  according  to 
the  laws  of  optics.     This  part  is  susceptible  of  calculation  ;  and  ac- 
cordingly it  is  considered  as  a  branch  of  applied  mathematics.  Aerial 
perspective  consists  in  the  exact  adaptation  of  light  and  distinctness, 
according  to  the  distance  of  the  object.     It  cannot  be  subjected  to 
mathematical  calculation,  on  account  of  the  imperfection  of  theoreti- 
cal and  practical  photometry. 

515.  When  a  body  appears  to  us  to  move,  what  we  perceive  is 
not  its  real  but  its  apparent  motion.     A  body  which  is  situated  in 
the  axis  of  the  eye,  and  which  advances  or  recedes  in  the  direction 
of  this  line,  appears  to  us  at  rest,  provided  it  is  not  so  near  that  we 
can  perceive  the  change  of  its  apparant  magnitude  and  distance.    In 
other  cases  it  is  always  the  motion  of  the  image  on  the  retina  which 
we  perceive  ;  and  this  may  obviously  be  very  different,  from  the  real 
motion  of  the  object ;  for  when  the  eye  itself  is  in  motion,  the  images 
change  their  place  upon  the  retina,  while  the  objects  which  they  re- 
present, are  at  rest.     If  the  observer  is  not  sensible  of  his  own  mo- 
tion, he  is  apt  to  suppose  that  it  is  the  objects  themselves  which 
move.     If  the  eye  and  the  object  seen  are  both  in  motion  at  the 
same  time,  the  phenomena  become  more  complicated.     This  is  the 
case  with  the  apparent  course  of  the  planets  in  the  celestial  sphere. 

516.  It  is  evident  from  what  precedes,  that  all  optical  deceptions 
are  not  false  sensations,  but  false  judgments,  to  which  our  sensations 
often  give  rise.     Our  judgments  would  be  much  more  erroneous,  if 
our  sensations  themselves  could  deceive  us  and  present  false  images. 
This  is  what  sometimes  happens  in  nervous  disorders. 


CHAPTER  XLI. 

Reflection  of  Light  by  Mirrors,  or  First  Principles  of  Catoptrics. 

517.  PROPERLY  speaking,  all  polished  surfaces  reflect  after  the 
manner  of  mirrors  ;  even  when  we  look  obliquely  at  a  polished  sur- 


240  Optics. 

face,  we  see  certain  images  similar  to  those  which  are  represented 
in  a  mirror ;  but,  for  the  most  part,  they  are  indistinct.  Among 
solid  bodies  there  are  only  certain  simple  metals,  and  certain  amal- 
gams of  metals,  which  are  susceptible  of  a  perfect  polish.  Looking- 
glasses  form  no  exception  to  this  remark ;  for  it  is  properly  the  amal- 
gam of  mercury  and  zinc,  with  which  the  posterior  surface  is  coated 
that  produces  the  effect. 

518.  Looking-glasses  have  indeed  rendered  metallic  mirrors  use- 
less for  ordinary  purposes  ;  but  they  cannot  be  employed  for  exact 
optical  experiments,  because  they  produce  a  double  reflection  on  the 
two  surfaces  of  the  glass,  and  also  because  the  light  which  arrives  at 
the  posterior  surface  is  itself  refracted  twice  in  the  glass,  and  conse- 
quently the  phenomena  which  we  observe,  are  not  produced   by 
reflection  alone.  These  inconveniences  are  the  more  to  be  regretted, 
since  it  is  difficult  to  prepare  a  good  composition  for  metallic  mirrors. 

519.  Among  the  infinitely  varied  forms  which  may  be  given  to 
the  surfaces  of  mirrors,  there  are  only  two  of  which  it  is  important 
to  speak  particularly  ;  these  are  plane  and  spherical  mirrors.    Under 
the  latter  denomination  we  include  all  those  which  are  portions  of  a 
sphere  polished  on  the  concave  or  convex   surface.     Various  at- 
tempts have  been  made,  without  success,  since  the  time  of  Descartes, 
to  polish  mirrors  of  elliptic  and  parabolic  curvatures,  &c.     Hut  inde- 
pendently of  the  almost  insurmountable  difficulties  to  be  met  with  in 
constructing  them,  it  is  demonstrated  by  theory  that  they  would  be 
inferior  to  spherical  mirrors.     Conical  and  cylindrical  mirrors  are 
used  only  for  purposes  of  amusement. 


Fundamental  Law  of  Catoptrics. 

520.  All  the  luminous  phenomena  which  are  produced  by  meant; 
of  mirrors,  though  infinitely  varied,  rest  upon  one  extremely  simple 
law,  which  is  the  following  ; 

If  a  ray  of  light  HA  (fig.  60)  falls  upon  any  surface  BAG, 
DAE,  or  FAG,  and  if  we  erect  at  thf  point  of  incidence  A,  the  //'/, 
AI  perpendicular  to  the  mirror,  and  suppose  a  plane  passing  thrc 
this  line  and  the  incident  ray,  the  reflected  ray  will  also  be  in 
plane,  and  will  make  unth  the  perpendicular  AI,  an  angle  IAK 
to  the  angle  1AH,  formed  by  the  incident  ray  with  this  same  pei 
dicular. 


Plane  Mirrors.  241 

In  a  word,  the  incident  ray  and  reflected  ray  have,  with  respect 
to  the  perpendicular  ./2/and  the  mirror,  an  opposite  but  symmetrical 
position.  We  call  dl  the  perpendicular,  Liff  the  angle  cf  inci- 
dence, and  L/1K  the  angle  of  reflection.  If  a  ray  falls  perpendicu- 
larly upon  a  mirror,  the  angle  of  incidence,  tnd  consequently  that  of 
reflection  are  zero,  that  is,  the  ray  is  reflected  back  upon  itself. 

The  exactness  of  this  law  may  be  proved  by  experiment  in  differ- 
ent ways  ;  and,  in  general,  it  is  sufficient  for  this  purpose,  to  render 
visible  the  directions  of  the  incident  and  reflected  rays.  One  of  the 
most  simple  methods  of  doing  this,  is  to  cause  the  light  of  the  sun, 
alter  passing  through  a  very  small  opening:,  to  fall  upon  the  surface 
of  a  mirror  in  a  darkened  room,  where  the  particles  of  dust  floating 
in  the  air,  are  illuminated  by  the  incident  and  reflected  light. 


Plane  Mirrors. 

521.  The  well  known  phenomena  of  the  plane  mirror  are  very 
easily  explained  by  the  above  law.  LPT  .#/?  (jig.  61)  be  the  profile 
of  such  a  mirror  ;  C  a  radiating  point  situated  before  its  surface  ; 
draw  the  line  CD  perpendicular  to  the  mirror,  and  produce  it  mak- 
ing /IE  equal  to  DC.  If  now  any  ray  CF  coming  from  C,  falls 
upon  the  mirror,  we  have  only  to  draw  through  E  and  F  the  line 
EFG,  in  order  to  find  the  position  of  the  reflected  ray  ;  for  the 
equality  of  the  triangles  FDC,  FDE,  being  easily  demonstrated,  it 
follows  that  the  angles  DFC,  DFE,  are  equal ;  but  BFG  and 
DFE  are  equal,  being  vertical  angles ;  consequently,  DFC  and 
BFG  are  also  equal  ;  therefore,  according  to  the  law  of  catoptrics, 
FG  is  the  reflected  ray. 

We  see,  therefore,  that  all  the  rays  coming  from  C  are  reflected 
by  the  mirror,  in  such  a  manner  that  their  directions  pass  through 
the  same  point  E.  Consequently  an  eye  placed  before  the  mirror, 
in  such  a  position  as  to  receive  one  of  these  reflected  rays,  must  see 
in  E  a  representation  of  the  point  (/'.  But  what  has  been  demon- 
strated for  the  point  C  is  applicable  to  every  other  point.  Thus  we 
see  why  the  image  of  an  object,  given  by  a  plane  mirror,  always  ap- 
pears to  be  situated  behind  the  mirror  at  a  distance  equal  to  the  dis- 
tance of  the  object. 

Elem.  31 


242  Optics. 


Spherical  Mirrors. 

522.  Let  J1DB,  (jig.  62)  be  the  profile  of  a  spherical  mirror, 
and  C  the  centre  of  the  spherical  surface  of  which  this  mirror  is  a 
part.  We  call  this  point  the  geometric  centre  ;  and  D  which  is 
the  middle  point  of  the  segment  itself,  is  called  the  optic  centre.  A 
straight  line  drawn  indefinitely  through  C  and  D.  represents  the 
axis.  CD  is  the  radius  of  the  mirror,  and  DA,  DB,  are  the  semi- 
apertures.  If  the  interior  surface  is  polished,  the  mirror  is  concave  of 
converging  ;  if  the  exterior  surface  is  polished,  it  is  convex  or  diverg- 


Phenomena  produced  by  Concave  Mirrors. 

523.  If  we  direct  the  axis  of  a  concave  mirror  towards  the  sun, 
all  the  rays  which  meet  its  surface  will  be  concentrated  by  reflection 


*  When  we  do  not  aim  at  rigorous  exactness,  we  mav  produce  the 
phenomena  of  the  two  kinds  of  mirrors,  with  glass  mirrors;  but  then 
the  luminous  ray  traversing  the  anterior  surface  is  reflected  before  it 
reaches  the  posterior  surface  which  reflects  it.  Accordingly,  in  this 
case,  we  must  not  judge  as  before,  of  the  properties  of  the  mirror, 
by  a  mere  inspection  of  its  anterior  surface  ;  it  is  only  when  the  two 
faces  are  parallel,  or  rather  when  they  are  of  concentric  curvature, 
a  point  very  difficult  to  be  effected  with  exactness,  that  the  mirror 
can  be  called  converging  when  its  anterior  surface  is  concave,  and 
diverging,  when  its  anterior  surface  is  convex.  But  if,  on  the  con- 
trary, the  two  faces,  as  is  usually  the  case,  are  of  different  forms  and 
curvatures,  we  call  that  a  convex  or  diverging  mirror,  of  which  the 
borders  are  thinner  than  the  middle  ;  and  that  a  concave  or  con- 
verging mirror  of  which  the  borders  are  thicker  than  the  middle  ; 
the  anterior  surface  being  indifferently,  either  plane,  convex,  or  con- 
cave, because  it  contributes  to  the  reflection  only  in  a  minuto  de- 
gree, compared  with  the  other  surface  which  is  coated  Those  who 
are  acquainted  with  the  effect  of  these  two  kinds  of  mirrors,  know 
that  the  converging  mirror  magnifies  an  object  placed  between  C  and 
D>  (fig-  6s)  and  that  the  diverging  mirror  diminishes  it.  Regard  bring 
paid  to  this  observation,  we  can  perform  the  experiments  indicated 
in  articles  522,  523,  524,  525,  with  glass  mirrors. 


Phenomena  produced  by  Concave  Mirrors.  243 

into  a  small  space  F,  which  is  exactly  in  the  middle  between  C  and 
D.  Hence  there  is  at  this  point  not  only  a  dazzling  light,  but  also 
an  intense  heat,  which  may  be  increased  almost  indefinitely.  For 
this  reason  we  call  this  point  the  focus  of  the  mirror,  and  DF  its 
focal  distance.  To  render  this  effect  as  great  as  possible,  the 
mirror  should  be  very  large,  and  its  focal  distance  less  than  the 
breadth  of  the  mirror,  or  at  least  not  exceeding  it ;  for  the  greater 
the  focal  distance  is,  compared  with  the  surface  of  the  mirror,  the 
less  will  be  the  effect  produced  at  the  focus.  A  body  which  we 
wish  to  expose  to  the  heat  of  the  focus  of  a  mirror,  should  be  smaller 
than  this  focal  space,  in  order  to  be  surrounded  on  all  sides  by  the 
heat  which  is  there  concentrated.  A  concave  mirror  fitted  for  this 
purpose,  is  called  a  burning  mirror. 

524.  If  we  place  a  flame  in  the  focus  of  a  concave  mirror,  all  the 
light  which  falls  upon  the  surface  will  be  reflected  nearly  parallel  to 
the  axis.     And  as  parallel  light  always  preserves  an  equal  force, 
except  when  it  is  weakened  by  the  absorbing  power  of  the  medium 
through  which  it  passes,  we  may  thus  propagate  a  vivid  light  to  a 
considerable  distance. 

525.  The  images  of  objects  represented  by  a  concave  mirror, 
present  phenomena  much  more  varied  than  those  of  a  plane  mirror. 
If  we  place  a  lighted  candle  before  the  mirror,  in  a  dark  room,  the 
following  phenomena  become  perfectly  manifest. 

(1.)  If  the  flame  is  between  the  focus  and  the  mirror,  we  see  a 
vertical  magnified  image  of  it,  which  appears  to  be  a  little  further 
behind  the  mirror,  than  the  flame  is  before  it.  As  we  bring  the 
light  towards  the  focus,  the  image  increases  in  magnitude  and  dis- 
tance. 

(2.)  If  we  place  it  at  the  focus,  the  image  disappears,  and  we  see 
only  the  luminous  reflection  described  in  the  preceding  article,  which 
consists  almost  entirely  of  parallel  rays. 

(3.)  If  we  place  the  light  beyond  the  focus,  we  no  longer  per- 
ceive its  image  in  the  mirror ;  but  when  it  is  at  a  certain  distance,  a 
large  inverted  image  is  painted  on  a  white  screen  or  wall  opposite 
the  mirror  ;  if  we  remove  the  light  still  further,  this  image  is  brought 
nearer  and  becomes  smaller.  When  the  distance  of  the  flame  be- 
comes double  the  focal  distance,  the  image  and  object  coincide,  since 
in  this  case,  the  flame  is  at  the  centre  of  curvature  of  the  mirror. 
If  we  remove  it  still  further,  the  image  now  smaller  than  the  flame, 
approaches  the  focus,  and  at  length,  when  the  light  is  removed  to  an 


244  Optics. 

infinite  cPstanee,  the  image  coincides  with  the  focus.  Hence  we  sec 
that,  in  a  burning  mirror,  "the  violent  heat  which  we  observe  at  the 
focus,  is  produced  by  an  image  of  the  sun  which  is  there  represented. 
526.  It  is  only  by  the  calculus  that  we  can  give  a  complete  expla- 
nation of  these  phenomena.  Yet  there  is  a  very  simple  and  ingen- 
ious method  of  determining  by  geometrical  construction,  what  the 
phenomenon  must  be  in  any  given  case  ;  it  supposes,  however,  some 
mathematical  analyses  of  which  we  shall  here  state  only  the  results. 
At  the  end  of  the  chapter  rigorous  demonstrations  may  be  found. 
The  propositions  are  the  following. 

(1.)  Each  ray  directed  parallel  to  the  axis,  is  reflected  to  the 
focus. 

(2.).  All  the  rays  which  come  from  any  point  in  the  direction  of 
the  axis,  or  very  nearly  in  this  direction,  are  reflected  in  such  a  man- 
ner that  their  directions  all  cut  each  other  in  one  point,  and  conse- 
quently produce  there  an  image  of  the  radiating  point ;  but  this 
image  is  sometimes  before  and  sometimes  behind  the  mirror  ;  it  may 
even  be  at  an  infinite  distance,  and  then  the  reflected  rays  are  par- 
allel. 

The  consequence  of  this  principle  is,  that  if  we  only  know  th« 
direction,  which  two  rays  from  the  same  point  take,  when  reflected, 
we  know  also  the  direction  of  all  the  rest. 

(3.)  When  several  points  are  at  equal  distances  from  the  mirror, 
their  images  are  also  equally  distant  from  the  mirror.  It  is  in  conse- 
quence of  this,  that,  when  we  place  an  object  before  a  mirror,  the 
reflected  rays  must  always  produce  an  image,  either  before  or  be- 
hind the  mirror. 

527.  If  we  admit  the  truth  of  these  propositions,  it  may  be  de- 
monstrated that  we  can,  in  each  case,  determine  all  the  conditions 
of  the  formation  of  images,  if  we  know  only  the  two  r;n  s  which  come 
from  the  extreme  points  of  an  object.  For  this  purpose,  let  ACB, 
(Jigs.  63,  64,  65)  be  the  section  of  a  mirror  CD,  its  axis,  and  E 
its  focus.  Let  the  straight  line  FG,  perpendicular  to  the  axis,  re- 
present the  radiating  object.  This  line  should  be  neither  less  nor 
greater  than  the  altitude  of  the  mirror,  and  should  extend  to  equal 
distances  on  each  side  of  the  axis.  From  the  highest  point  F  of 
this  object,  we  draw  two  rays,  FA  and  FC,  to  the  mirror  ;  F.I  par- 
allel to  the  axis,  is  reflected  to  the  focus  E ;  FC  directed  to  the 
optic  centre  C  of  the  mirror,  will  be  reflected  towards  the  Imu-st 
point  of  the  object,  according  to  the  law  of  catoptrics.  Produce  the 


Phenomena  produced  by  Convex  Mirrors.  245 

directions  of  these  two  reflected  rays  till  they  cut  each  other.  The 
point  of  their  intersection/,  is  the  image  of  the  point  F  of  the  object. 
If  we  draw  from  this  point  /  a  line  fg,  passing  perpendicularly 
through  the  axis,  making  g  h  equal  to  fh,  this  line  will  represent  the 
image  reflected  by  the  mirror  in  the  circumstances  supposed. 

Figure  63  represents  the  case  in  which  the  object  FG  is  within 
the  focal  distance  CE.  The  two  reflected  rays  J1E  and  CG,  which 
come  originally  from  JP,  here  diverge,  and  we  consequently  produce 
them  behind  the  mirror  to  find  their  point  of  intersection/  which  is 
the  image  of  the  point  F,  as  also  fg,  which  is  the  image  of  the 
entire  object.  This  is  an  illustration  of  the  phenomenon  mentioned 
in  article  526,  (1.) 

In  figure  64,  the  object  FG  is  itself  at  the  focus  E.  Here  the 
two  reflected  rays  AE  and  CG  become  parallel ;  for  since,  accord- 
ing to  the  remark  made  at  the  end  of  article  525,  the  mirror  must 
have  only  a  small  curvature  to  produce  a  distinct  image,  we  may 
consider  CAFE  as  a  parallelogram  ;  but  then  CAEG  is  also  a  par- 
allelogram, since  CA  and  EG  are  equal.  In  this  case,  there  will 
be  no  image  formed,  or  rather  we  may  say,  there  will  be  one  at  an 
infinite  distance  behind  or  before  the  mirror.  This  explains  the  sec- 
ond phenomenon  mentioned  in  article  525. 

Figure  65  represents  the  object  FG,  beyond  the  focal  distance. 
The  two  reflected  rays  J\f  and  Cf  are  here  sensibly  convergent, 
and  produced  sufficiently  far,  they  would  cut  each  other  below  the 
axis  in/  so  that  in  this  case  the  mirror  produces  in  the  air  an  image 
fg  of  the  object.  This  explains  the  third  phenomenon  of  article  525. 

We  may  treat  in  the  same  manner  all  other  cases  which  present 
themselves.  The  reader  may  here  examine  the  changes  which 
take  place  in  the  last  of  these  phenomena,  according  as  the  object  is 
placed  between  the  geometric  centre  and  the  focus,  or  at  the  geo- 
metric centre.  These  are  mentioned  at  the  end  of  article  525. 


Phenomena  produced  by  Convex  Mirrors. 

528.  The  phenomena  presented  by  a  convex  mirror,  when  the 
light  of  an  object  falls  upon  it,  are  much  more  simple  than  the  pre- 
ceding. Whenever  we  place  an  object  before  the  mirror,  we  per- 
ceive an  image  smaller  than  the  object  itself,  and  situated  vertically 
behind  the  mirror.  When  we  direct  the  axis  of  a  convex  mirror 


246  Optics. 

towards  the  sun,  instead  of  concentrating,  it  disperses  the  light.  But 
it  may  be  proved  as  well  by  experiment  as  by  calculation,  that  the 
small  image  of  the  sun,  from  which  this  dispersion  of  light  proceeds, 
is  placed  at  an  equal  distance  between  the  optic  and  geometric  cen- 
tre, and  consequently  behind  the  mirror.  Hence  we  call  this  point 
the  negative  focus  of  the  mirror,  and  its  distance  from  the  mirror,  the 
negative  focal  distance. 

529.  The  theoretical  propositions  stated  in  article  526,  may  be 
applied  to  convex  mirrors,  as  well  as  to  concave  ones  ;  only  the  ex- 
pression of  the  first  must  be  changed  thus.     A  ray  parallel  to  the 
axis  must  be  reflected  as  if  it  came  from  the  negative  focus.     With 
this  modification  the  construction  described  in  article  527,  may  serve 
also  for  convex  mirrors. 

Let  ACB  (Jig.  66)  be  the  profile  of  such  a  mirror,  ED  its  axis, 
and  E  its  negative  focus.  Let  FG  be  the  object ;  from  F  draw 
the  ray  FA,  parallel  to  the  axis.  It  will  be  reflected  in  the  direction 
AK)  as  if  it  came  from  E.  The  ray  FC  is  reflected  toward 
G.  The  two  reflected  rays  obviously  diverge,  and  consequently  we 
must  produce  them  behind  the  mirror  in  order  to  find  their  point  of 
intersection/,  and  represent  the  entire  image  fg  of  the  object  FG. 

530.  It  is  necessary  to  know  perfectly  the  focal  distance  of  a 
spherical  mirror,  when  we  wish  to  employ  it  in  exact  experiments. 

In  the  case  of  a  concave  mirror,  there  are  several  ways  of  deter- 
mining the  focal  distance.  For  example,  we  present  the  mirror  to 
the  rays  of  the  sun  in  such  a  manner  that  their  direction  shall  be  par- 
allel to  its  axis,  and  measure  the  distance  of  the  image  from  the  mir- 
ror. Or  we  cut  a  piece  of  paper  of  the  shape  and  size  of  the  mir- 
ror, draw  a  diameter  across  it,  and  in  the  diameter  make  two  round 
holes  at  equal  distances  from  the  centre.  This  we  place  upon  the 
mirror  and  then  expose  it  to  the  light  of  the  sun.  The  rays  re- 
flected through  the  two  holes  will  converge  ;  we  find  their  point  of 
meeting  and  measure  its  distance  from  the  mirror. 

A  third  method  will  be  given  among  the  additions  at  the  end  of 
the  chapter. 

For  a  convex  mirror,  we  can  only  make  use  of  the  second  method. 
When  we  have  attached  the  paper  to  the  mirror,  we  perceive  that 
reflected  rays  diverge,  and  we  are  to  seek  the  points  where  they  are 
at  a  distance  from  each  other  equal  to  double  that  which  they  had 
upon  the  mirror.  We  measure  the  distance  of  these  points  from  the 
mirror,  and  thus  we  have  the  negative  focal  distance. 


Mathematical  Additions.  247 


Mathematical  Additions. 

531.  The  propositions  enunciated  in  articles  526  and  529,  are  not 
rigorously  but  conditionally  exact.     This  condition  is,  that  they  ap- 
proach the  more  nearly  to  the  truth,  in  proportion  as  the  extent  of 
the  mirror  is  less  compared  with  its  focal  distance,  or  with  the  radius 
of  the  sphere  upon  which  the  mirror  is  constructed.     Yet  it  may  be 
proved  by  a  more  extended  analysis,  that  the  inaccuracy  of  which 
we  speak,  is  scarcely  perceptible,  even  when  the  spherical  segment 
which  forms  the  mirror,  is  one  of  several  degrees.     In  order  that 
mirrors  may  afford  clear  and  distinct  images,  the  diameter  of  the 
mirror  should  never,  at  most,  exceed  half  the  focal  distance,  and  in 
certain  cases  it  ought  to  be  much  less. 

This  remark  justifies  the  approximations  to  which  we  shall  con- 
fine ourselves  in  the  following  demonstrations. 

532.  Theorem.     A  luminous  ray  EA  (Jig.  67),  which  falls  par- 
allel to  the  axis  upon  a  concave  mirror,  is  reflected  between  the 
optic  centre  D,  and  the  geometric  centre  C,  and  the  nearer  to  the 
focus  F ,  the  nearer  it  passes  to  the  axis. 

Demonstration.  If  we  draw  from  A  to  C,  the  straight  line  AC, 
it  will  be  a  radius  of  the  sphere  ADB,  and  consequently  perpen- 
dicular at  A,  to  the  surface  of  the  mirror.  If  we  take  the  angle 
CAP  =  CAE,  AE  being  the  incident  ray,  AF  will  the  reflected 
ray. 

If  now  we  consider  the  triangle  AFC,  it  will  be  readily  perceived 
that  AF  =  FC,  since  the  angles  FAC  and  FCA  are  equal ;  for 
each  is  equal  to  the  angle  EAC ;  the  first  by  construction,  and  the 
second,  because  AE  being  parallel  to  FC,  FCA  and  CAE  are 
alternate  internal  angles.  Now  if  we  had  AF=  DF,  we  should  also 
have  DF  equal  to  FC,  and  thus  the  point  F  would  be  exactly  in 
the  middle  of  the  line  DC.  This  does  not  exactly  take  place  for 
all  rays ;  but  the  difference  between  the  shortest  line  DF,  and  the 
longest  AF,  evidently  becomes  smaller,  according  as  AD  is  small 
compared  with  DF  or  DC.  If  the  arc  AD,  or  the  angle  AFD, 
comprehends  only  a  few  degrees,  we  may,  without  material  error, 
suppose  DF  =  AF.  Then  DF  =  FC.  Thus  the  first  proposi- 
tion of  article  526  is  demonstrated. 

533.  Theorem.     In  a  convex  mirror  ADB  (fig.  68),  the  ray  EA 
parallel  to  the  axis,  mil  be  reflected  in  the  direction  AH,  as  if  it  had 

• 


248  Optics. 

come  from  the  middle  of  the  radius  CD.  The  demonstration  is 
similar  to  the  preceding.  The  line  CAG  is  perpendicular  to  the 
arc  ADB  in  A.  If,  therefore,  we  make  GAH  =  GAE,  AH  is 
the  reflected  ray,  which,  being  produced,  will  cut  the  axis  in  F. 
Now  in  the  triangle  CAF,  we  have  CF  =  AF,  since  CAP  =  ACF; 
for  CAF  =  HAG,  being  vertical  angles,  and  ACF  =  GAE,  be- 
ing internal  external  angles.  But  HAG  —  GAE  on  account 
of  the  fundamental  law  of  reflection  ;  consequently  CAF  must  be 
equal  to  ACF.  But  FA  and  FD  are  not  strictly  equal ;  they  only 
approach  to  an  equality  under  certain  conditions ;  that  is,  CF  is 
more  nearly  equal  to  FD,  according  as  the  incident  ray  passes 
nearer  the  axis.  Thus  the  principle  supposed  in  article  529  is  de- 
monstrated. 

534.  Problem.  In  the  axis  ED  of  the  spherical  mirror  ADB 
(fig.  69),  there  is  a  radiating  point  E.  A  ray  EA  emanating  from 
this  point,  falls  upon  the  mirror  in  A  and  is  reflected  towards  F.  It 
is  proposed  to  find  an  equation  between  the  focal  distance  of  the  mir- 
ror =  ±  DC  =  p,  the  distance  if  the  luminous  point  DE  =  a,  and 
the  distance  DF  =  «,  at  which  the  reflected  ray  cuts  the  tn  /.v. 

Solution.  Let  C  be  the  geometric  centre  of  the  mirror  ;  CA 
will  be  perpendicular  to  its  surface  at  A.  Consequently,  from  the 
law  of  catoptrics,  CAF  =  CAE.  But  AFD,  being  the  exterior 
angle  is  equal  to  the  two  opposite  interior  angles,  FAC,  ACF. 
Whence  FAC  =  AFD  — ACF;  also  CAE  =  ACF—AHC: 
consequently,  AFD  —  ACF  rr  ACF  —  AEC,  or 

2  ACF  =  AFD  -\-AEC. 

Moreover,  it  is  shown  in  treatises  on  trigonometry,  that  when  a  right- 
angled  triangle  has  one  of  the  acute  angles  very  small,  this  angle  is 
nearly  proportional  to  the  side  opposite  divided  by  the  side  adjacent, 
and  this  proposition  approaches  so  much  nearer  the  truth  according 
as  the  opposite  side  is  less.  Now,  if  we  wish  to  obtain  distinct 
images,  we  must  regard  the  arc  AD  as  very  small  compared  with 
DF,  DC,  and  DE ;  we  may.  therefore,  consider  it  as  a  straight 
line  perpendicular  to  the  axis  DE,  and  consequently  the  triangles 
ADF,ADC,  ADE,  as  right-angled  triangles,  having  very  small  an- 
gles at  the  base,  F,  C,  E.  Consequently, 


Mathematical  Additions.  249 


the  angle  A  CF  is  proportional  to  —  — 

L/C 


••••«' 


If  in  the  above  equation  2^CP  i=  AEC  +  .tfFD,  we  substitute 
for  these  angles  the  values  which  are  proportional  to  them,  we  shall 
have 

2  AD  _  AD    ,AD 
~D~C    ~  DE  ~*~  ~DF' 

or,  dividing  the  whole  by  J1D, 

JL  •  •  _L  +  _L 

D'    "     LE  "*"  DF' 

Finally,  if  we  substitute  2p  for  DC,  a  for  DE,  and  «  for  DF,  we 

shall  have 


which  is  the  equation  sought  between  /?,  «,  and  ". 

535.  Remark.  The  formula  which  we  have  investigated  has  a  very 
extensive  application  ;  and  we  shall  show  that  all  possible  phenomena 
which  take  place  in  mirrors  and  spherical  glasses,  may  be  repre- 
sented by  it,  and  that  consequently  it  may  be  regirded  as  the  basis 
of  all  optical  calculations.  It  is  desirable  to  give  a  simple  enuncia- 
tion to  a  proposition  so  important. 

For  this  purpose,  we  observe  that  it  is  common  to  call  the  quo- 
tient arising  from  dividing  unity  by  any  quantity,  the  reciprocal  value 

of  that  quantity.     Thus  -  is  the  reciprocal  value  of  p,  and  so  of  the 

rest.     A  quantity  and  its  reciprocal  value  have  such  a  ratio  to  each 
other,  that   if  we  know  one,  we  always  find  the  other  by  divid- 

ing unity  by  that  which  is  known.     Thus  1  divided  by  -  gives  p. 

As  we  find  one  of  these  values  so  easily  from  the  other,  we  may  in- 
differently consider  one  or  the  other  as  the  quantity  known  or  sought. 
It  is  advantageous,  therefore,  to  keep  the  above  equation  in  its  pre- 
sent form,  and  not  to  eliminate  the  divisors,  because  it  would  lose 
much  of  its  simplicity  and  utility. 
Elern.  32 


250  Optics. 

If  we  call  DE  =  a,  and  DF  =  «,  the  two  distances  afwhich  the 
rays  meet,  the  above  formula  expresses  the  following  theorem  ; 

The  reciprocal  valve  of  the  focal  distance  is  equal  to  the  sum  of  the 
reciprocal  values  of  the  two  distances  at  which  the  rays  meet. 

536.  Additions.  (I.)  It  is  an  essential  property  of  every  algebraic 
formula,  that  it  is  not  only  applicable  to  the  particular  case  taken  as  the 
basis  of  the  calculation,  but  that  it  serves  also  for  all  imaginable  cases 
of  the  same  kind.  We  must  remark,  however,  that  when  we  apply 
it  to  other  cases,  it  is  sometimes  necessary  to  change  the  sign  of  one 
of  the  quantities.  In  the  case  upon  which  this  formula  is  founded,  we 
have  considered  all  the  quantities  p,  a,  a,  as  positive,  by  supposing 
them  placed  with  respect  to  one  another,  as  represented  in  figure 
69.  But  if  one  of  these  lines,  in  another  case,  has  a  contrary  situa- 
tion, we  must  give  it  the  negative  sign.  With  this  modification,  our 
formula  is  applicable  to  all  imaginable  cases,  in  which  an  incident 
ray  Ed  (fg-  69)  cuts  the  axis  in  any  point  E.  As  long  as  the 
point  E  is  before  the  mirror,  the  quantity  a  has  the  positive  sign. 
But  if  the  ray  does  not  come  from  a  point  of  the  axis,  but,  on  the 
contrary,  is  directed  towards  one  of  these  points,  as  GA  (fig-  70) 
is  directed  towards  E,  the  distance  DE,  which  was  before  the 
mirror  in  figure  69,  is  now  behind  it ;  so  that  we  must  represent 
DE  by —  a,  and  the  formula  will  then  become  for  a  concave  mirror 

-  = 1 — .     If.  moreover,  the  mirror  in  question  is  convex, 

p  a        « 

the  ray  and  the  focal  distance  have  a  position  opposite  to  those 
represented  in  figures  69,  70;  we  must,  therefore,  represent  the 
focal  distance  by  —  p.  In  this  case,  when  the  radiant  point  is 
before  the  mirror,  as  in  figure  69,  a  remains  positive  ;  and  the  for- 
mula is :=  -  -f-  _.  But  if  the  intersection  of  the  axis  CD,  by  the 

p       a       a 

incident  ray,  were  behind  the  mirror  as  in  figure  70,  a  would  be  also 

negative,  and  consequently  we  should  have = 1-  -.    And 

p  a       a 

so  on. 

What  has  been  said  can  be  applied  only  to  quantities  considered 
in  a  particular  case,  as  given  quantities.  But  it  is  obvious  that  each 
of  the  three  quantities  may  be  considered  as  unknown,  when  the 
other  two  are  known.  Thus  the  formula  serves  generally  to  find  one 
of  these  quantities  by  means  of  the  other  two,  and  it  determines,  at 
the  same  time,  the  sign  to  be  given  to  it. 


Mathematical  Addition*.  251 

(2.)  Since  AD  (fig.  69)  is  entirely  eliminated  from  the  calcula- 
tion, this  is  a  proof  that  the  magnitude  of  this  arc  does  not  sensibly 
influence  the  position  of  the  point  F,  where  the  reflected  ray  cuts 
the  axis,  provided  that  AD  is  in  general  a  very  small  arc,  as  the 
whole  calculation  supposes.  It  follows  then,  that  not  only  the  ray 
EA,  but  all  the  rays  coming  from  the  same  point  E,  will  unite  after 
reflection,  nearly  in  the  same  point  F,  and  there  produce  an  image 
of  the  point  J3,  which  will  be  visible  to  an  eye  placed  in  such  a 
manner  as  to  receive,  at  some  distance,  the  divergent  rays  coining 
from  F. 

(3.)  As  the  formula  applies  to  all  positions  of  the  point  E  in  the 
axis,  it  is  demonstrated  that  for  each  radiant  point  situated  in  the 
axis,  there  is  always  produced  by  reflection  a  new  image  of  this 
point,  situated  in  this  same  axis.  This  image  is  before  the  mirror,  if 
the  formula  gives  a  positive  value  for  « ;  behind  it,  when  this  value 
is  negative  ;  and  at  an  infinite  distance  if  this  value  is  infinite,  or 

which  amounts  to  the  same  thing,  if  -  =  0.     This  is  the  case  with 

a 

a  concave  mirror,  when  we  suppose  a  =  p,  for  then  we  have 

1        1         1  1 

-  = -,  that  is,  -  =.  0. 

p        p         a  a 

Hence  we  see  when  the  rays  proceed  from  the  focus,  they  are 
reflected  parallel  to  the  axis ;  in  other  words,  their  point  of  union  is 
at  an  infinite  distance. 

537.  Problem.  To  determine  the  circumstances  of  reflection 
when  the  radiant  point  is  without  the  axis,  but  at  a  small  distance 
from  it. 

Solution.  Let  G  (Jig.  71)  be  a  radiant  point  near  the  axis. 
Draw  the  straight  line  GCH  through  the  geometrical  centre,  and 
produce  it  till  it  meets  the  mirror.  It  is  evident  that  this  line  may 
be  considered  absolutely  as  an  axis,  since  KDB  is  spherical.  If 
therefore,  a  ray  GK  falls  upon  the  mirror,  and  is  reflected  towards 
GL,  by  making  HG  =  a,  and  HL  =  «,  we  shall  have  as  above 

l-  =  -  +  I 

p        a        a 

And  all  the  consequences  which  we  have  deduced  with  respect  to 
the  axis  are  true  with  respect  to  the  line  GH.  Hence  it  follows 
that  each  radiant  point  situated  in  the  line  GH,  produces  an  image 
somewhere  in  this  line.  This  image  may  be  according  to  circum- 


252  Optics. 

stances,  sometimes  bercre  the  mirror,  sometimes  behind  it,  and  some- 
times at  an  infinite  distanr.  . 

Thus  the  second  supposition  of  article  526  is  demonstrated. 

538.  Additions.  (1.)  As  we  suppose  the  size  of  the  mirror  very 
small  compared  with  the  focal  distance,  and  the  radiant  point  G  to 
be  near  the  axis,  it  is  evident  that  all  the  lines  drawn  from  G  to  the 
mirror,  will  be  nearly  equal  in  length.  The  same  is  true  of  all  the 
lines  drawn  from  L  to  the  mirror.  Hence  it  follows  that  the  formula 


\\'\\  vary  but  little  from  the  truth,  even  when  we  do  not  measure  the 
distances  a  and  «,  but  substitute  their  perpendicular  distance  from 
the  mirror.  It  follows  also  that  if  several  radiant  points  are  situated 
below  G,  at  equal  distances  from  the  mirror,  their  images  above  L 
would  also  all  be  at  equal  distances  from  its  surface.  For  since  a  is 
the  same  for  all  those  points,  the  formula  would  give  equal  values 
for  a.  Thus  the  third  supposition  of  article  526  is  sufficiently  evi- 
dent. 

(2.)  If  we  represent  the  object,  as  we  have  done  above,  by  a 
strni':ht  line  perpendicular  to  the  axis,  the  image  will  also  he  a 
straight  line  perpendicular  to  the  axis.  Then  we  may  call  a  the  dis- 
t  <e  of  the  entire  object,  and  not  merely  that  of  a  radiant  point; 
and,  in  like  manner,  «  will  be  the  distance  of  the  entire  image.  For 

this  value  of  the   letters  a  and  «,  the  formula  -  =  -  -4-  -,  remains 

p        a        a 

always  exact. 

539.  Remarks.  (1.)  The  last  observation  furnishes  a  convenient 
method  of  finding  the  focal  distance  of  a  concave  mirror.  We 
place  before  the  mirror  the  flame  of  a  candle  at  such  a  distance,  that 
a  distinct  image  may  be  formed  on  a  white  screen  or  wall  properly 
placed.  Then  we  measure  the  distance  of  the  image  and  object 
from  the  mirror,  and  obtain  a  and  «,  from  which  p  may  be  found 
by  means  of  the  formula. 

(2.)  All  the  calculations  in  optics  become  difficult  and  complicat- 
ed, when  we  undertake  to  make  them  with  strict  accuracy.  But  in 
practice  this  exactness  is  not  necessary,  when  the  question  relates  to 
instruments  which  are  to  afford  very  distinct  images.  For,  in  order 
to  obtain  this  distinctness,  mirrors  must  have  a  very  small  surface 
compared  with  their  focal  distance,  which  consideration  justifies  all 


First  Principles  of  Dioptrics.  253 

ihe  approximations  we  have  employed  ;  and  as  to  instruments  where 
•great  precision  is  not  sought,  this  circumstance  of  itseli  justifies 
them. 


CHAPTER  XLII. 

Refraction  of  Light  in  Transparent  Bodies  ;  or  First  Principles  of 
Dioptrics. 

540.  ALL  aeriform   fluirls,  most  liquids,  and  many  solid  bodies, 
are  transparent.    Perhaps  there  is  no  one  which  may  not  be  travers- 
ed by  light  to  a  certain  degree  ;  since  gold   itself,  which  in   large 
masses  is  opaque  and  dense,  appears  to  have  a  kind  of  transparency 
when  reduced  to  thin  leaves.     Most  transparent  bodies  pt.rmit  listht 
to  traverse  them,  without  altering  it,  that  is,  without  changing  the 
colour,  which  it  had  before  penetrating  them.    But  many  only  trans- 
mit certain  colours,  and  hence  appear  coloured.     There  are   ;tlso 
bodies  which  reflect  one  colour  and  transmit  another,  as  gold  leaf, 
tincture  of  turnsol,  &tc. 

54 1.  Li  order  to  perfect  transparency,  the  surfaces  both  of  solids 
and  liquids  must  be  perfectly  polished.     This  condition  is  fulfilled 
naturally  in  liquids,   by   the  simple  effect  of  gravity  which  renders 
their  surfaces  perfectly  plane.     It  is  also  fulfilled  to  a  certain  degree, 
in  crystallized  bodies.     In  gener.il,  however,  the  assistance  of  art  is 
necessary  for  a  sufficiently  exact  polish.     When  a  transparent  body 
is  not  polished,  it  suffers  the  light  to  pass;  but  then  it  disperses  it 
irregularly  in 'all  directions,  and  we  cannot  see  distinctly  through  it. 
Among  transparent  bodies,  the  greater  number  refract   light  simply  ; 
that  is,  the  collections  of  luminous  rays  are  not  separated  in  traversing 
them  ;  but  there  are  other  bodies  which  separate  the  rays  into  two 
distinct  portions.     To  this  class  belong  all  crystallized  bodies,  the 
primitive  form  of  which  is  neither  a  cube  nor  a  regular  octaedron. 
This  phenomenon  is  called  double  refraction.     We  shall  here  con- 
sider only  simple  refraction  as  being  the  most  common  and  most  sim- 
ple in  its  theory. 


254  Optics. 


Laiv  of  Dioptrics. 

542.  All  the  phenomena  observed  by  means  of  transparent  bodies 
which  refract  light  simply,  may  be  explained  by  the  following 
law. 

When  a  luminous  ray  passes  obliquely  from  one  transparent  me- 
dium into  another,  it  undergoes  a  refaction,  or  deviates  from  its 
primitive  direction.  If,  through  the  point  of  incidence,  where  the  ray 
meets  the  second  medium,  we  sup/Jose  a  line  drawn  perpendicular  to 
the  refracting  surface,  the  refracted  ray  mil  approach  this  perpen- 
dicular, when  the  mei'ium  which  it  enters  is  more  dense  than  that 
which  it  leaves  ;  and,  on  the  contrary,  will  diverge  from  it,  when  it 
is  more  rare. 

In  order  to  demonstrate  this  law,  let  A  (fig.  72)  be  the  point 
where  the  ray  passes  from  one  medium  into  nnother  ;  let  the  surface 
which  separates  the  two  media  be  plane  as  BC,  convex  as  DE,  or 
concave  as  FG.  Suppose  the  rarer  medium  to  be  above  and  the 
denser  one  below  ;  let  the  incident  ray  be  HA.  If  at  A  we  erect 
the  perpendicular  IAK  at  the  point  of  incidence,  and  suppose  a 
plane  passing  through  IAK and  AH,  the  refracted  ray  will  also  be 
in  this  plane,  but  in  such  a  manner  that  the  angle  KAL,  which  is 
in  the  denser  medium,  will  be  smaller  than  the  angle  HAI,  which  is 
in  the  rarer.  With  A  for  a  centre  and  a  radius  taken  at  pleasure, 
we  describe  the  circle  HILK.  From  the  points  H  and  L  where 
the  incident  and  refracted  rays  cut  the  circumference,  we  draw  the 
lines  HM  and  LJV,  perpendicular  to  the  vertical  /  IK.  It  is  proved 
by  experiment,  that  these  two  lines  HM  and  LN,  have  always  inva- 
riable ratios,  for  all  directions  of  incidence,  the  two  media  in  which 
the  light  moves  remaining  the  same. 

In  a  right-angled  triangle,  of  which  the  hypothenuse  is  supposed 
equal  to  unity,  the  two  other  sides,  expressed  in  numbers,  that  is,  in 
parts  of  the  hypothenuse,  are  called  the  sines  of  the  opposite  angles. 
Since  the  magnitude  of  the  radius  AK  is  arbitrary,  we  may  here 
consider  it  as  unity  ;  then  HJM  will  be  the  sine  of  the  angle  HAI, 
or  the  sine  of  incidt  nee,  and  UV  the  sine  of  LAK,  or  the  sine  of 
refraction ;  and  the  law  of  refraction  may  be  briefly  expressed  as 
follows ; 

When  a  ray  passes  from  one  medium  into  another,  it  is  rffracted 
in  such  a  manner  that  the  sine  of  incidence  and  the  sine  of  refraction 


IMW  of  Dioptrics.  255 

«re  i)i  a  constant  ratio  to  each  other.  This  ratio  is  called  the  ratio 
of  refraction.  It  is  customary  to  call  the  angles  .EM/ and  LdK,  by 
the  name  of  the  media  in  which  they  are  situated  ;  as  the  angle  in 
the  air,  in  the  water,  &tc. 

543.  Among  experiments  made  in  conformity  to  this  law,  the 
one  most  easily  compreh  nded,  if  not  the  most  exact,  is  the  follow- 
ing.    A  glass  cube  ABCD  (fig.  73)  is  placed  upon  two  boards 
joined  at  right  angles,  as  represented  by  EC  and  CF. .  They  should 
be  longer  than  the  side  of  the  cube.     If  we  expose  this  apparatus  to 
the  light  of  the  sun,  so  that  the  luminous  ray  shall  fall  in  the  direc- 
tion GH,  this  ray  GH  will  be  refracted  to  UK  in  the  glass  ;  but, 
on  the  outside  it  will  pursue  its  primitive  direction  to  F .    The  shad- 
ow of  the  board  CE  will,  therefore,  reach  to  /Tin  the  glass,  and  to  F 
without  it.     Now  it  we  draw  through  H  the  incident  vertical  LHM, 
it  will  be  easily  seen  that  the  angle  FH.M  is  equal  to  the  angle  in 
the  air  GHL,  and  that  KHM  is  the  angle  in  the  glass.  If  we  meas- 
ure the  length  of  the  shadow  within  and  without  the  glass,  we  can 
determine  the  two  angles,  either  by  calculation  or  by  construction. 
Then  if  we  cause  the  light  to  fall  under  different  angles,  and  trace  a 
figure  for  each  case,  we  may  mark  the  sines  of  the  angles,  and  find 
their  ratio  by  means  of  an  exact  scale ;  and  these  will  verify  the 
law  enunciated. 

The  experiments  may  be  made  more  accurately  with  a  glass 
prism  ;  but  they  could  not,  in  this  case,  be  comprehended  without 
a  more  perfect  knowledge  of  the  theory  than  we  can  here  suppose. 

544.  Before  the  middle  of  the  seventeenth  century,  it  was  believed 
that  the  angles  themselves,  and  not  their  sines,  were  in  a  constant  ratio 
to  each  other.     Snellius  of  Holland  corrected  this  idea,  and  made 
known  the  exact  principle.     Yet  when  the  angles  HA1  and  LJlKt 
(jw.  72)  are  very  small,  we  may  without  inconvenience  take  this 
ratio  for  the  angles,  since  they  are  sensibly  proportional  to  their  sines  ; 
and  since  we  never  make  use  of  great  angles  in  exact  dioptric  instru- 
ments, we  may  admit  the  ratio  of  the  angles  as  constant  in  the  cal- 
culations relative  to  these  instruments. 

545.  With  respect  to  the  particulars  of  the  preceding  law,  we 
are  to  observe  the  following  circumstances. 

( ! .)  If  a  ray  falls  perpendicularly,  as  AI  (Jig.  72)  it  passes  with- 
out being  refracted  ;  in  all  other  cases  it  is  refracted,  and  the  quan- 
tity of  refraction  is  greater  in  proportion  to  the  obliquity  of  the  inci- 
dence. 


256  Optics. 

(2.)  A  ray  of  light  takes  the  same  direction  between  two  media, 
whether  it  be  considered  as  entering  or  emerging,  other  circum- 
stances being  the  same  ;  that  is,  if  LA  w-re  an  incident  ray,  AH 
would  be  the  refracted  ray. 

(3  )  At  each  refraction  there  is  alvvnys  a  reflection  at  the  polished 
surface,  whether  the  ray  passes  from  the  denser  into  the  rarer  medi- 
um, or  the  reverse  ;  that  is,  if  the  ray  is  broken  at  A,  a  part  is 
reflected  according  to  the  law  of  catoptrics,  and  the  other  part  is 
refracted  according  to  the  law  of  dioptrics.  The  more  obliquely 
the  ray  falls,  the  greater  is  the  portion  reflected,  and  the  less,  conse- 
quently, the  portion  refracted  ;  for  all  polished  surfaces  reflect  light 
much  more  strongly  in  oblique  directions,  than  when  it  falls  perpen- 
dicularly upon  them.  Even  when  the  ray  is  directed  from  a  denser 
to  a  rarer  medium,  there  is  a  limit  beyond  which  there  can  be  no 
angle  of  refraction  in  the  other  medium,  since  the  sine  of  this  angle 
would  be  greater  than  unity,  which  is  impossible  ;  and  then  all  the 
light  is  reflected.  In  a  glass  filled  with  water,  it  will  be  easily  seen 
that  not  only  the  upper,  but  also  the  lower  surface,  r«  fleets,  and  that 
the  latter  reflects  much  more  than  the  former,  especially  if  we  look 
very  obliquely. 

(4.)  Moreover,  at  each  refraction,  a  remarkable  change  takes 
place  in  light.  We  shall  merely  mention  it  here,  and  afterwards 
attend  to  it  more  particularly.  After  refraction,  the  luminous  ray  is 
no  longer  a  simple  straight  line,  but  enlarges  into  a  pyramidal  form, 
and  each  point  of  its  breadth  exhibits  a  different  colour.  Still  this 
expansion  is  very  slight  for  a  single  refracted  ray,  especially  near 
the  refracting  surface.  We  shall  neglect  this  circumstance  in  the 
present  chapter,  and  represent  the  refracted  ray  as  a  single  straight 
line. 

(5.)  Dense  bodies  refract  light  more  strongly,  other  things  being 
the  same,  than  those  which  are  more  rare.  Yet  the  refracting  power 
does  not  depend  upon  the  density  alone,  but  also  upon  the  chemical 
composition.  Thus  we  have  observed  that  combustible  bodies  re- 
fract light  more  strongly  than  incombustible.  Our  knowledge  upon 
this  point,  however,  is  so  slight,  that  we  can  only  ascertain  the  re- 
fracting power  of  each  body  by  direct  experiment.* 

*  M.  Arago  and  myself,  some  years  ago,  made  a  variety  of  experi- 
ments on  this  subject  with  the  repea'inn  circle.  We  ascertained 
that  hydrogen  gives  to  oils,  rosins,  and  other  substances  which  we 


PJienomcna  which  Depend  upon  Refraction.  257 

(6.)  The  tnost  interesting  ratios  of  refraction  are  those  which 
exist  hetween  air  and  glass,  and  between  air  and  water.  That  be- 
tween air  and  common  glass  is  nearly  3  : 2,  or  more  exactly  17:11. 
Between  air  and  English  crown  glass,  it  is  1,55  :  1  ;  between  air 
and  flint  glass  1,58  :  1.  Between  air  and  water,  it  is  nearly  4  :  3, 


General  Phenomena  which  depend  upon  the  Refraction  of  Light. 

546  If  light  were  neither  refracted  nor  reflected  by  transparent 
bodies,  those  which  are  perfectly  transparent  and  colourless  would 
be  invisible.  We  can  see  them  only  in  consequence  of  the  reflec- 
tion which  takes  place  at  their  surfaces,  and  the  difference  of  direc- 
tion which  refraction  produces  in  the- light  which  traverses  them. 
Thus  we  can  even  distinguish  two  colourless  fluids,  which  exist 
together  in  the  same  vessel  without  mixing  ;  such  as  oil  and  water, 
or  ether  and  water,  &ic.  The  air  is  invisible  in  small  masses, 
because  the  refractions  and  reflections  are  insensible.  When  a 
visible  body  is  in  a  different  transparent  medium  from  that  where 
the  eye  is,  its  apparent  position,  in  most  cases,  undergoes  a  change 
by  the  refraction  of  light. 

Let  A  (Jig.  74)  be  a  visible  point  at  the  bottom  of  a  vessel  full 
of  water  BAC.  A  ray  AD  which  falls  vertically  upon  the  surface 
of  the  water,  penetrates  it  without  being  refracted  ;  but  this  is  the 
only  direction  in  which  we  see  the  point  in  its  actual  place.  The 
ray  AE  which  enters  the  surface  of  the  water  under  an  acute  angle, 
is  refracted  in  the  air  and  diverges  still  farther  from  the  incident  per- 
pendicular, drawn  through  E.  It  continues,  therefore,  through  the 
air  as  if  it  came  from  a  more  elevated  point  a ;  and  then  an  eye 
which  is  in  the  prolongation  of  the  ray  EF,  must  see  the  point  A  in 
the  direction  EF,  that  is,  in  a. 

What  has  been  said  of  A  is  applicable  to  all  other  points  at  the 

call  combustible,  their  great  refracting  power.  We  also  ascertained 
that  the  refracting  power  of  the  ingredients,  is  in  the  ratio  of  their 
masses,  when  the  state  of  aggregation  is  not  changed  ;  so  that  by 
means  of  this  law,  we  can  calculate  beforehand  with  considerable 
accuracy,  the  refracting  force  of  bodies,  and  deduce  from  it  some 
inferences  respecting  the  nature  and  proportions  of  their  constituent 
principles. 

Elem.  38 


258  Of  tics. 

.bottom  of  the  vessel.  Thus  all  this  part  must  appear  to  be  raised  ta 
B  a  C.  If  a  straight  staff  GHA  is  immersed  in  water,  the  part 
below  the  surface  will  appear  broken,  because  each  of  the  points 
which  compose  it  must  appear  more  elevated  than  it  really  is. 

If  the  eye  were  at  *#,  and  the  object  observed  were  in  the  line 
EF,  it  would  not  be  seen  in  its  true  direction,  but  in  the  prolonga- 
tion of  the  line  rfE. 

We  find  ourselves  in  such  a  situation  with  respect  to  the  heavenly 
bodies  ;  and  astronomers  have  long  since  observed  that  the  heavenly 
bodies  which  are  not  in  the  zenith,  appear  more  distant  from  the 
horizon  than  they  really  are.  This  is  called  astronomical  refraction. 


Particular  Phenomena  which  are  produced  by  means  of  Polished 
Glasses. 

547.  Polished  glasses  give  rise  to  phenomena  too  important  to 
be  neglected  here.  There  are  two  kinds  of  glasses  which  we  shall 
examine  ;  those  whose  faces  are  plane  and  parallel,  and  those  whose 
faces  are  portions  of  a  sphere.  In  one  of  the  following  chapters  we 
shall  speak  of  glasses  whose  surfaces  are  plane,  but  inclined  to  each 
other  ;  that  is,  of  prismatic  glasses. 


Plane  Glasses  with  Parallel  Faces. 

548.  Let  ABCD  (jig.  75)  be  the  profile  of  a  glass  of  this  kind, 
and  EF  a  luminous  ray  falling  upon  its  anterior  surface.  At  the 
point  of  incidence  F,  erect  the  perpendicular  GH.  The  ray  will 
be  refracted  in  the  glass  at  F,  and  will  take  the  direction  Fl.  At  /, 
the  point  of  emergence,  erect  a  second  perpendicular  KL,  which 
will  be  parallel  to  the  first.  The  ray  will  again  be  refracted  in  the 
air  at  this  point,  and  will  take  the  direction  IM.  We  easily  see  that 
1M  is  parallel  to  EF ;  for  since  the  two  angles  in  the  glass,  HFI 
and  FIR:  are  equal,  the  angles  in  the  air,  EFG  and  L1M  must  also 
be  equal. 

By  refraction  in  such  glasses  all  tho  emergent  rays  remain  parallel 
to  the  incident  rays ;  thus  we  must  see  through  such  a  glass  pre- 
cisely as  we  should  see,  if  no  glass  were  there.  Only  when  we 
look  very  obliquely,  the  objects  must  change  place  a  little,  yet  with- 


Spherical  Glasses  or  Lenses.  259 

out  changing  their  magnitude  or  respective  situations.  In  all  other 
cases,  the  direction  of  the  ray  is  so  little  changed,  that  we  may  con- 
sider its  refraction  as  nothing. 


Spherical  Glasses  or  Lenses. 

549  The  different  kinds  of  microscopes  and  telescopes  are  in- 
struments indispensable  to  the  philosopher.  They  consist  of  glasses 
the  faces  of  which  are  portions  of  a  sphere.  In  order  to  under- 
stand the  effect  of  compound  optical  instruments,  it  is  necessary 
first  to  consider  the  properties  of  the  simple  glasses  of  which  they 
are  formed. 

550.  Although  the  form  of  spherical  glasses  may  be  varied  much 
more  than  that  of  mirrors,  we  may  nevertheless,  in  considering  their 
essential  properties,  arrange  them  in  two  classes  ;  convex  or  converg- 
ing glasses,  and  concave  or  diverging  glasses.     Each  of  these  two 
classes  are  subdivided  as  follows.     The  converging  glasses  are, 

(i.)  Double  convex,  as  in  figure  76.  The  form  of  this  glass  is 
lenticular,  and  for  this  reason  we  usually  call  it  a  lens  ;  and  the  term 
applies  not  only  to  glasses  of  this  kind ;  but  also  to  all  spherical 
glasses,  particularly  the  smallest. 

(2.)  Plano-convex,  as  in  figure  77. 

(3.)  Concavo-convex,  as  in  figure  78.  The  word  convex  must 
be  placed  last,  to  denote  that  the  convexity  here  is  greater  than  the 
concavity.  We  call  such  a  glass  a  meniscus,  on  account  of  the  form 
of  its  profile. 

Diverging  glasses  are, 

(1.)  Double  concave,  as  in  figure  79. 

(2.)  Plano-concave,  as  in  figure  80. 

(3.)  Convex-concave,  as  in  figure  81.  It  is  usual  to  apply  the 
term  meniscus  to  this  lens  also,  the  form  of  which  requires  that  we 
put  the  word  concave  last. 

551.  With  respect  to  all  these  glasses,  we  make  the  following 
general  remarks ; 

(1.)  Their  faces  are  made  to  take  the  spherical  form,  for  the 
same  reason  that  determined  us  to  give  this  form  to  mirrors. 

(2.)  The  phrase  radius  of  curvature  imports  the  semidiameter  of 
the  sphere  of  which  the  surface  in  question  is  a  portion. 


260  Optics. 

(3.)  In  order  to  see  distinctly  through  these  glasses,  it  is  neces- 
sary that  their  surfaces,  like  those  of  mirrors,  should  not  be  large 
portions  of  a  sphere.  We  may  lay  this  down  as  the  limit,  of  their 
extent,  that  the  arc  of  the  segment  should  be  at  most  equal  to  only 
half  the  radius  of  curvature. 

(4.)  In  the  middle  of  a  glass  of  this  kind  (Jig.  76,  81),  there  is 
a  point  C,  where  die  two  opposite  faces  are  parallel.  This  point  is 
called  the  t.ptic  centre  of  the  glass.  A  line  DE  drawn  through  this 
point,  perpendicular  to  the  two  faces,  is  called  the  axis  of  the  glass. 
In  this  line  are  situated  the  geometric  centres  of  the  two  faces,  that 
is,  the  centres  F  and  G  of  the  two  spheres  of  the  surface  of  which 
these  faces  are  portions. 

When  the  optic  centre  and  the  point  of  intersection  of  the  axis, 
are  exactly  in  the  middle  of  the  exterior  surface,  we  say  that  the 
glass  is  exactly  centred.  This  is  an  essential  quality  for  optical  pur- 
poses. The  equal  thickness  of  the  exterior  circumference  indicates 
this  property,  but  not  with  all  the  exactness  necessary.  The  most 
certain  indication  is  when  the  objects  do  not  change  their  apparent 
position,  if  we  move  the  glass  circularly  in  a  plane  perpendicular 
to  its  axis. 

When  we  wish  to  make  use  of  these  glasses,  it  is  common  to  cover 
a  portion  of  their  borders  with  an  opaque  ring,  and  the  interior  diam- 
eter of  this  ring  is  called  the  aperture  of  the  glass. 

(5.)  The  anterior  surface  of  the  glass  is  always  that  which  is 
turned  towards  the  object,  and  the  posterior  surface,  that  which  is 
turned  towards  the  eye. 

(6.)  All  converging  glasses  produce  phenomena  essentially  similar. 
The  same  is  true  of  all  diverging  glasses  compared  with  one  another. 
The  advantages  of  the  different  kinds,  depend  upon  circumstances, 
and  these  advantages  cannot  be  determined  or  even  understood 
clearly  without  a  knowledge  of  mathematics.  In  general,  double  con- 
vex or  double  concave  glasses  are  perferred,  especially  if  their  cur- 
vatures are  symmetrical,  because  they  admit  of  the  greatest  apertures. 

(7.)  Experience  has  proved  that  common  mirror  glass  of  a  slightly 
greenish  colour,  is  the  best  for  all  optical  instruments.  English  flint 
glass  is  used  only  for  particular  purposes. 


Phenomena  produced  by  Converging  Glasses.  26  J 


Phenomena  produced  by  Converging  Glasses. 

552.  When  we  expose  a  converging  glass  to  the  sun,  and  receive 
the  light  transmitted  through  it,  on  a  white  surface,  this  light  is  col- 
lected in  a  certain  space,  the  extent  of  which  varies  with  the  posi- 
tion of  the  surface.     If  it  is  first  very  near  the  glass,  and  we  gradu- 
ally remove  it,  the  luminous  space  becomes  smaller.     Hence  the 
term  converging.    At  length  we  reach  a  point  where  the  light  occu- 
pies the  least  possible  space,  and  beyond  that  it  becomes  divergent. 
This  point  is  railed  the  focus,  and  its  distance  from  the  nearest  sur- 
face of  the  glass  is  called  the  focal  distance.     If  we  turn  the  glass 
and  expose  the  other  surface,  the  same  phenomenon  takes  place.   A 
converging  glass  has  therefore  two  foci,  and  they  are  equally  distant 
from  the  two  surfaces,  if  these  have  the  same  radius.     In  glasses 
whose    surfaces  are  not  symmetrical,  especially   meniscuses,  these 
distances  differ,  but  by  a  quantity  scarcely  perceptible. 

553.  A  burning  glass  is  a  convex  glass  of  considerable  extent,  as 
two  or  three  feet,  and  having  its  focal  distance  equal  to  the  aperture  ; 
or  at  least  very  little  exceeding  it.     The  effects  of  such  a  glass  be- 
come rnor*3  intense,  according  to  the  extent  of  the  surface,  and  the 
smallness  of  the  space  in  which  the  rays  are  concentrated.     If  the 
focus  is  very  distant,  this  space  is  extended,  and  consequently  the 
effect  is  small.     In  this  case  it  is  common  to  place  at  some  distance 
another  convex  glass,  called  a  collector,  which  causes  the  rays  to 
converge  into  a  smaller  space.     The  effects  of  burning  glasses  are 
as  remarkable  as  those  of  burning  mirrors. 

554.  The  focal  distance  of  a  symmetrically  double  convex  glass, 
is  equal  to  the  radius  of  either  surface,  or  rather  to  if-  of  this  radius. 
For  a  plano-convex  glass,  it  is  equal  to  double  the  radius,  or  more 
exactly  to  y .     We  shall  demonstrate  in  the  mathematical  additions 
at  the  end  of  the  chapter,  what  is  its  value  with  respect  to  the  radii 
of  glasses  not  symmetrical. 

555.  The  other  properties  of  convex  glasses  have  the  closest  re- 
semblance to  those  of  converging  mirrors,  and  may  be  exhibited  with 
even  greater  facility,  by  means  of  a  lighted  candle  in  a  dark  room. 

(1.)  When  we  place  the  candle  before  the  glass  and  within  the 
focal  distance,  the  eye  placed  on  the  other  side  of  the  glass  sees  the 
image  magnified,  erect,  and  very  distant,  the  magnitude  and  distance 
increasing  as  we  remove  the  light. 


262  Optics. 

(2.)  If  we  place  the  candle  at  the  focus,  we  do  not  see  any  dis- 
tinct image  of  it,  but  only  a  vivid  light  which  consists  principally  of 
parallel  rays,  and  continues  behind  the  lens,  so  as  to  illuminate  dis- 
tant objects. 

(3.)  If  we  place  the  candle  at  a  certain  distance  beyond  the  focu.n, 
we  see  a  magnified  and  inverted  image  on  the  opposite  wall.  If  we 
continue  to  move  it  still  farther,  this  image  approaches  the  posterior 
focus  of  the  glass,  and  becomes  smaller.  If  the  flame  is  placed  at 
double  the  focal  distance,  the  image  is  at  the  same  distance,  and  has 
the  same  dimensions  as  the  flame  itself.  If  we  remove  the  flame 
still  more,  the  image  approaches  and  becomes  smaller,  and  if  the 
object  be  very  distant,  it  falls  at  length  in  the  focus.  Accordingly 
the  caustic  space  within  which  a  burning  glass  burns,  is  nothing  more 
than  the  small  image  of  the  sun  formed  at  the  focus.  If  wj  do  not 
receive  the  image  formed  under  the  preceding  circumstances,  upon 
a  white  screen  or  ground  glass,  an  eye  placed  at  a  proper  dis- 
tance will  see  it  formed  in  the  air.  But  for  reasons  which  are  easily 
conceived,  the  imagination  represents  it  not  where  it  really  is,  but 
in  the  glass  itself,  or  rather  on  the  opposite  face. 

556.  So  far  as  we  can  understand  these  phenomena  without  the 
aid  of  mathematics,  it  will  be  perceived  that  they  do  not  differ  essen- 
tially from  those  described  in  articles  526,  527.     In  this  case,  as 
before,  if  we  take  some  principles  for  granted,  we  can  determine  by 
an  easy  construction,  what  phenomena  will  be  exhibited  in  a  given 
case.     To  facilitate  this  construction,  it  is  to  be  remarked  that  a  ray 
which  passes  through  the  optic  centre  C  (jig*  76,  81),  must  be  con- 
sidered as  not  refracted.     For  the  rays  which  make  small  angles 
with  the  axis,  this  is  clear  from  the  position  of  the  surfaces  at  C ; 
for,  since  they  are  here  parallel,  the  ray  which  passes  through  must 
be  refracted  as  in  a  plane  glass  with  parallel  faces. 

557.  After  these  preliminary  observations,  figures  82,  84,  do  not 
require  much  explanation.     JIB  is  the  section  of  a  convex  glass ; 
C  is  the  optic  centre,  DQ  the  axis,  D  the  anterior  focus,  E  the 
posterior  focus,  FHG  the  radiant  object. 

(1.)  In  figure  82  the  object  is  at  the  middle  of  the  anterior  focal 
distance  DC ;  from  its  most  elevated  point  F,  a  ray  FA  falls  upon 
the  glass  parallel  to  the  axis,  and  is  refracted  towards  the  pos- 
terior focus  E.  A  second  ray  FCI  passes  through  the  optic  centre 
without  being  refracted.  The  rays  J1E  and  CI  diverge  after  their 
passage  ;  and  if  produced  in  the  opposite  direction,  they  will  cut  each 


.Phenomena  produced  by  Converging  Glasses.  263 

other  in/;  accordingly  all  the  rays  from  F,  appear  to  come  from 
this  point/.  An  eye  placed  behind  the  glass  sees  in/  the  image  of 
the  point  F ;  and  instead  of  the  object  GF,  it  will  see  the  image  fg. 

(2.)  Figure  83  represents  the  object  FG  placed  at  the  anterior 
focus  D ;  the  parallel  ray  FA  is  refracted  towards  E,  the  ray  FC 
passes  without  being  refracted  ;  but  as  the  lines  CE  and  FA  are 
equal  and  parallel,  since  CE  =  CD  =  FA,  it  follows  that  CFAE 
is  a  parallelogram,  and  the  rays  AE.  C/,  are  parallel  after  passing 
through  the  glass.  The  same  is  true  of  all  the  rays  which  proceed 
from  the  point  F. 

(3.)  In  figure  84  the  object  FG  is  placed  beyond  the  anterior 
focal  distance  DC  ;  the  parallel  ray  FA  is  refracted  toward  E'  the 
ray  FC/  passes  without  being  refracted  ;  AE  and  Cf  converge 
after  their  passage,  and  their  prolongations  cut  each  other  in/;  all 
the  rays  coming  from  F  unite  at  this  point,  and  an  inverted  image  of 
the  object  is  formed  in/g. 

After  these  explanations  it  will  not  be  difficult  to  construct  the 
figures  peculiar  to  the  third  case,  according  as  the  object  is  at  the 
centre  of  curvature,  or  at  a  much  greater  distance  before  the  glass. 

558.  A  very  extensive  and  varied  use  may  be  made  of  a  single 
converging  glass. 

(1.)  The  effect  of  the  common  kind  of  spectacles  is  explained  by 
the  construction  of  figure  82.  These  are  used  to  remedy  one  of 
the  defects  of  sight  common  to  aged  persons,  by  which  the  distance 
of  distinct  vision  becomes  so  great  that  the  objects  with  which  we 
are  more  immediately  concerned,  are  seen  only  in  a  confused  man- 
ner. By  the  help  of  spectacles  we  are  able  to  see  objects  at  a  con- 
venient distance.  To  determine  the  proper  focal  distance  for  spec- 
tacles, regard  must  be  paid  ;  1.  To  the  distance  of  distinct  vision  ; 
2.  To  the  distance  at  which  we  are  accustomed  to  read,  or  to  place 
small  objects  in  order  to  see  them  conveniently.  For  this  reason 
different  eyes  require  different  focal  distances.  There  are  specta- 
cles which  have  their  focus  at  the  distance  of  from  16  to  20  inches. 
It  is  prudent  to  begin  with  using  these,  and  to  pass  very  slowly  to 
shorter  focal  distances,  in  order  to  preserve  the  sight  as  long  as  pos- 
sible. 

An  important  remark  suggests  itself,  which  is  applicable  to  all 
kinds  of  simple  glasses  and  compound  optical  instruments.  When 
we  look  through  a  glass,  the  eye  experiences,  almost  always,  an  ex- 
traordinary tension,  which  may  become  very  injurious  to  this  organ. 


264  Optics. 

We  have  already  observed  that  we  do  not  perceive  immediately  the 
distance  of  object",  but  use  our  judgment  in  estimating  it  according 
to  circumstances.  When  we  see  by  the  aid  of  glasses,  we  want  almost 
all  the  means  which  serve  to  guide  us  in  estimating  the  magnitude 
and  distance  of  the  image,  and  commonly  the  imagination  phices  the 
image  at  a  false  distance.  The  eye  accordingly  adapts  itself  to 
this  false  distance  ;  and  from  this  contradiction  between  the  true  dis- 
tance from  which  the  refracted  rays  come,  and  that  which  is  sup- 
posed by  the  imagination,  there  must  result  a  certain  state  unnatural 
to  the  eye.  This  observation  explains  many  singular  phenomena 
which  result  from  the  use  of  glasses,  and  also  the  difference  of  the 
judgment  which  different  persons  form  respecting  the  magnitude  and 
distance  of  objects  seen  through  an  optical  instrument ;  but,  on  ac- 
count of  the  limits  which  we  have  proposed  to  ourselves,  we  cannot 
treat  this  subject  more  at  length ;  and  shall  only  observe,  that  in 
order  to  avoid  injuring  the  sight  by  using  glasses,  we  must  first  learn 
to  see  with  the  aid  of  these  instruments. 

559.  (2.)  The  effects  of  simple  magnifying  glasses  depend  upon 
the  same  principles  ;  we  shall  easily  be  convinced  of  this,  if  we  con- 
sider that  in  figure  82  the  image  fg  is  greater  and  more  distant 
when  the  focal  distance  is  smaller,  and  consequently,  that  a  glass 
magnifies  the  more,  as  its  focal  distance  is  less.  Magnifying  glusses 
whose  focal  distance  is  from  half  an  inch  to  several  inches,  are  called 
magnifiers.  When  this  distance  is  less  than  half  an  inch,  they  are 
called  simple  microscopes,  or  microscopic  lenses. 

As  it  is  always  necessary  that  the  image  should  be  at  the  distance 
of  distinct  vision,  that  is,  about  8  inches  before  the  glass,  it  is  obvious 
from  a  mere  inspection  of  figure  82,  that  the  object  FG  must  always 
be  very  near  the  locus  D,  if  the  image  fg  is  to  be  removed  from 
the  glass  16  times  its  real  distance  or  more.  In  using  the  micro- 
scope, the  object  must  be  nearly  in  the  for  us.  According  to  this 
remark  it  is  easy  to  estimate  the  magnifying  power  of  a  microscope. 
For,  on  account  of  the  similarity  of  the  triangles  FHCandfh  C,  the 
object  FH  is  to  the  image  fh,  as  the  distance  of  the  object  HC,  is 
to  the  distance  of  the  image  h  C.  In  a  microscope,  HC  must  be 
only  a  little  smaller  than  the  focal  distance,  and  then  h  C  is  nearly  8 
inches  ;  thus  the  focal  distance  is  to  8  inches  as  unity  is  to  the  num- 
ber which  expresses  the  degree  of  enlargement  or  magnifying  power. 

500.  There  are  still  two  things  to  be  observed  with  respect  to  the 
magnifying  power. 


Phenomena  produced  by  Converging  Glasses.  265 

(1.)  Since  the  distance  of  distinct  vision  is  different  for  different 
eyes,  and  since,  moreover,  according  to  the  remark  made  in  the  pre- 
ceding article,  many  optical  illusions  combine  to  influence  our  esti- 
mate of  distance,  there  must  be  a  great  difference  in  the  magnitude 
ascribed  by  different  persons  to  the  same  object. 

(2.)  The  number  which  expresses  the  magnifying  power  accord- 
ing to  the  above  rule,  only  indicates  the  enlargement  of  the  diameter 
of  the  object.  To  obtain  the  enlargement  of  the  surface,  we  must 
square  this  number ;  to  obtain  that  of  the  volume,  we  must  cube  its 
A  microscope  whose  focal  distance  is  0,1  inch,  magnifies 

The  diameter  80  times, 

The  surface  6400  times, 

The  volume  512000  times. 

The  last  number  being  the  greatest,  is  that  commonly  used  to  indi- 
cate the  magnifying  power  of  a  microscope  ;  but  one  is  astonished 
that,  with  a  microscope  which  magnifies  half  a  million  of  times,  the 
diameter  should  only  appear  80  times  greater.  When  we  speak  of 
telescopes,  we  make  use  of  the  more  just  denomination,  which  ex- 
presses the  enlargement  of  the  diameter. 

561.  ( ?.)  The  effect  of  the  solar  microscope  may  be  explained 
by  figure  84.  If  we  place  FG  near  the  focus  D,  we  observe  that 
the  image  fg  is  removed  and  becomes  greater  ;  indeed  there  is  no 
degree  of  enlargement  which  this  image  is  not  capable  of  attaining. 
Consequently,  if  we  fix  a  small  glass  lens  to  the  shutter  of  a  very 
dark  room,  and  place  a  small  inverted  object  a  httle  beyond  the 
focal  distance  D-  ,  an  image  is  produced  at  a  certain  distance  be- 
hind .he  glass,  which  may  be  received  upon  a  white  screen  or  upon 
the  wall.  This  image  is  erect,  and  magnified,  but  it  is  obvious  that 
the  solar  light  which  colours  it  must  be  extremely  feeble,  and  the 
more  so,  according  as  the  image  is  greater,  if  the  object  is  not  illu- 
minated by  all  possible  means.  It  is  not  sufficient  to  illuminate  it 
directly  by  the  simple  light  of  the  sun  ;  this  light  must  be  collected 
nearly  to  a  point  by  means  of  a  convex  lens.  Then  the  size  of  the 
object  is  to  that  of  the  image,  as  CH  to  C  h,  that  is,  as  the  focal 
distance  of  the  lens  is  to  the  distance  of  the  screen.  The  solar 
microscope  has  this  advantage,  that  many  persons  can  see  the  image 
at  the  same  time  ;  and  also  that,  by  means  of  the  image,  the  figure 
of  tie  object  may  be  very  easily  delineated.  But  this  instrument 
does  not  admit  of  the  precision  that  belongs  to  the  simple  micro- 
Elem.  34 


266  Optics. 

scope ;  and  the  image  loses  in  distinctness  as  much  as  it  gains  iti 
magnitude,  \vhen  we  increase  the  distance  of  the  screen. 

5(32.  In  the  camera  obscura  the  images  of  distant  objects  are 
formed  by  a  converging  glass,  as  represented  in  figure  84,  and  are 
received  directly  upon  a  white  screen  or  upon  ground  glass  ;  or  they 
are  reflected  upward  or  downward  by  a  plane  mirror  placed  at  some 
distance  behind  the  glass,  and  making  with  it  an  angle  of  45° ;  so 
that  these  images  may  be  received  upon  a  horizontal  plane.  Land- 
scapes are  represented  in  a  very  agreeable  manner  in  the  camera 
obscura.  and  the  painter  may  advantageously  make  use  of  one. 

563.  We  shall  now  speak  of  some  instruments  composed  of  two 
converging  glasses,  but  otherwise  analogous  to  the  preceding. 

(3.)  The  camera  lucida  consists  of  a  quadrangular  box,  before 
which  is  placed  a  convex  glass  of  considerable  extent ;  behind  this, 
in  the  box,  is  a  plane  mirror,  placed  at  an  angle  of  45°,  which 
reflects  towards  the  cover  the  images  of  distant  objects,  which,  with- 
out it  would  have  been  painted  on  the  posterior  side.  In  front  of  the 
mirror  an  opening  is  made  to  which  a  second  converging  glass  is 
fitted,  and  through  which  we  view  the  object  as  with  a  simple  glass. 

(4.)  In  the  -magic  lantern,  there  are  two  large  convex  glasses  at 
a  little  distance  from  each  other.  Before  the  first,  as  in  FG  (Jig- 
62),  we  pass  a  figure  painted  upon  glass,  placed  within  the  focal 
distance,  and  illuminated  as  strongly  as  possible,  by  means  of  a  lamp 
and  a  reflecting  mirror  placed  behind.  As  the  figure  moves  within 
the  focal  distance  of  the  first  glass,  the  rays  after  their  passage 
through  this  glass,  continue  as  if  they  came  from  a  distant  and  in- 
verted image,  fg  (Jig.  82).  The  second  glass  must  be  placed 
beyond  the  first  JIB,  to  receive  the  transmitted  rays,  and  its  situa- 
tion must  be  such,  that  the  image  fg  shall  extend  a  little  beyond  its 
focal  distance.  Then  the  rays  are  refracted  in  this  second  glass,  so 
as  to  produce,  as  in  figure  81,  a  distant  image  fg,  which  may  be 
received  upon  a  white  screen.  This  image  is  erect,  because  that  of 
which  it  is  the  representation  is  inverted. 

(5.)  We  also  make  compound  magnifiers  consisting  of  two  convex 
glasses.  The  object  FG  is  placed  close  in  front  of  the  first  glass, 
and  the  rays  are  refracted  as  if  they  came  from  the  distant  image 
fg  (jig.  82.)  Close  behind  this  glass,  there  is  a  second  whose  an- 
terior focal  distance  extends  a  little  beyond  the  image  fg.  Through 
this  last  we  see  the  image  just  as  we  should  see  the  real  object  with 
the  simple  magnifier. 


Phenomena  produced  by  Diverging  Glasses.  267 

(6.)  There  are  instruments  which  consist  of  a  large  convex  glass, 
the  focal  distance  of  which  is  from  1|  to  2  feet;  these  are  pecu- 
liarly fitted  for  viewing  large  designs  in  perspective.  The  design 
being  well  illuminated,  must  be  placed  within  the  focal  distance,  but 
not  very  fer  from  the  focus.  Then  we  see  a  distant  magnified 
image,  rather  indistinct,  it  is  true,  but  for  this  very  reason  more  true 
to  nature. 

564.  It  is  obvious  from  what  precedes,  that  we  must  first  know 
the  focal  distance  of  a  converging  glass,  in  order  to  judge  of  its 
effects.  It  is  found  in  the  same  manner  as  that  of  mirrors. 

We  expose  the  glass  to  the  light  of  the  sun  or  moon,  and  measure 
the  distance  from  its  surface  to  the  image  produced  at  its  focus ; 
or  we  cover  the  glass  with  a  paper  circle  having  two  holes  made 
in  it,  and  ascertain  the  point  where  the  rays  which  pass  through 
these  holes  meet. 

Much  address  is  required  in  this  operation,  when  we  wish  to  deter- 
mine with  accuracy  very  near  or  very  distant  foci. 


Phenomena  produced  by  Diverging  Glasses. 

565.  The  phenomena  produced  by  diverging  glasses,  are  entirely 
analogous  to  those  which  we  obtain  by  means  of  convex  mirrors. 

If  we  direct  one  of  these  glasses  towards  the  sun,  and  receive 
upon  a  white  surface  the  light  which  it  transmits,  it  will  be  seen  that 
this  light  diverges  as  if  it  came  from  a  point  situated  in  the  concavity 
of  the  glass.  We  call  this  point  the  negative  focus,  and  its  distance 
from  the  anterior  surface,  the  negative  focal  distance.  If  we  reverse 
the  faces,  the  phenomena  are  the  same  ;  a  diverging  glass  has,  there- 
fore, two  negative  foci. 

The  luminous  rays  transmitted  through  a  diverging  glass,  form 
erect  images,  which  are  nearer  and  smaller  than  the  objects  them- 
selves. The  distance  of  the  object  occasions  no  other  modification 
in  these  phenomena,  except  to  make  the  image  appear  a  little  farther 
from  the  glass,  according  as  we  remove  the  object.  But  the  utmost 
limit  to  which  the  image  can  be  removed  is  the  anterior  focus,  and 
this  is  attained  when  the  object  is  at  a  very  great  distance. 

566.  Let  JIB  (fig.  85)  be  a  diverging  glass,  HD  its  axis,  and 
E  and  D  the  two  negative  foci.     Let  the  object  FG  be  at  H;  the 
phenomenon  which  takes  place  under  these  circumstances,  may  be 


268  Optics. 

determined  by  the  same  method  which  was  mir'e  use  of  for  mirrors 
HIV  i^ing  glasses.  From  the  most  elevated  point  .F,  draw 
tlie  ray  FA  parallel  to  the  axis ;  this  is  refracted  in  tne  direction 
AK,  and  then  seems  to  come  from  the  focus  K  ;  a  second  ray  FCL 
passes  through  the  optic  centre  without  being  refracted.  The  rays 
AK  and  CL  diverge,  therefore,  after  passing  through  the  glass,  as  if 
they  come  from  the  point  /;  this  then  is  ti  e  point  from  which  all 
the  rays  from  F  seem  to  come.  Thus  if  we  draw/g  perpendicular 
to  the  axis,  we  sh.;ll  have  the  magnitude  and  position  of  the  image 
seen  through  the  glass. 

;~>o7.  Jt  is  only  for  spectacles  that  we  employ  diverging  glasses 
separately.  They  bring  distant  objects  near,  so  that  short-sighted 
persons  can  see  them,  in  the  same  manner  as  convex  spectacles 
remove  those  which  are  too  near,  and  place  them  at  a  convenient 
distance  for  long-sighted  persons. 

The  remark  made  at  the  end  of  article  558,  applies  particularly 
to  concave  spectacles.  When  we  see,  for  example,  with  a  concave 
glass,  the  focal  distance  of  which  is  10  inches,  the  imagination  can- 
not easily  conceive  that  the  whole  extent  of  a  wide  prospect  is  con- 
tained in  a  space  of  10  inches  radius  ;  and  yet  all  the  images  which 
we  see,  are  actually  within  this  space.  For  this  reason,  the  imagi- 
nation always  removes  them  too  far,  and  thus  the  eye  experiences 
an  unnatural  tension.  When,  therefore,  we  are  obliged  to  use  this 
kind  of  spectacles,  we  should  begin  with  those  lenses  whose  focal 
distances  are  considerable,  and  pass  gradually  to  those  which  are 
shorter. 


Mathematical  Additions. 

568.  The  essential  theory  of  all  spherical  glasses,  or  at  least, 
what  is  necessary  for  understanding  all  the  phenomena  which  we 
have  described,  is  deduced  from  two  theorems,  one  of  which  relates 
to  the  refraction  of  a  ray  coming  from  any  point  of  the  axis ;  and 
the  o'h'T  to  the  refraction  of  a  ray  which  comes  from  a  point  situated 
very  near  the  axis. 

Figure  86  illustrates  the  first  of  these  principles.  Let  ADC  be 
the  upper  half  of  the  profile  of  a  converging  glass.  Let  D  be  the 
geometric  centre  of  the  anterior  surface  AC  ;  E  the  geometric  cen- 
tre of  the  posterior  surface.  Let  us  suppose  that  the  plane  of  the 


Mathematical  Additions.  269 

profile  passes  through  these  two  points.  A  line  FI  drawn  through 
D  and  E  must  be  perpendicular  to  A  and  B ;  consequently  it  will 
be  the  axis  of  the  glass.  From  the  point  F  of  the  axis,  the  ray 
FG  falls  upon  the  glass.  Through  G  and  D,  therefore,  draw  the 
normal  KGD ;  we  shall  find,  according  to  the  fundamental  law  of 
dioptrics,  that  the  refracted  ray  remains  in  the  plane  of  the  figure, 
and  that  it  makes  with  the  normal  GD  a  smaller  angle  in  the  glass 
than  in  the  air.  Let  GH  be  its  direction.  If  through  the  point  H 
where  it  reaches  the  posterior  surface,  we  draw  to  this  surface  the 
normal  EHL,  the  ray,  after  passing,  will  still  remain  in  the  plane  of 
the  figure  ;  but  it  will  diverge  from  HL  downward,  and  consequently 
will  cut  the  axis  in  some  point.  Let  /  be  this  point ;  then  the  prob- 
lem will  be  expressed  as  follows  ;  To  find  the  position  of  the  point  I, 
when  the  radii  of  the  surfaces,  the  positions  of  the  centres  and  those 
of  the  points  F  and  G,  are  known. 

If  we  wish  to  solve  this  problem  with  the  utmost  rigour,  we  must 
employ  long  and  complicated  calculations.  But  it  is  easy  to  obtain 
such  an  approximation  as  is  entirely  sufficient  for  all  practical  pur- 
poses. This  is  done  by  means  of  a  relation  which  exists  between 
the  two  radii  DA  and  EB,  the  two  distances  of  meeting  AF  and  Bl, 
and  the  ratio  of  refraction  which  we  suppose  to  be  known. 

The  circumstance  which  facilitates  this  approximation  is,  that  the 
arcs  must  have  a  very  slight  curvature,  in  order  that  the  glasses  may 
produce  distinct  images.  Consequently,  the  acute  angles  at  D  and 
E  are  always  very  small ;  but  then  the  acute  angles  F,  /,  G,  H,  are 
always  small  also,  and  the  arcs  GA,  HB,  may  be  considered  nearly 
as  lines  perpendicular  to  the  axis,  and  also  as  equal  and  parallel  lines, 
on  account  of  the  thinness  of  the  glass. 

569.  Problem.  Conformably  to  the  above  considerations,  let 
AD  =  /,  EB  =  g,  AF  =  a,  El  =  «,  and  let  the  ratio  of  refrac- 
tion between  air  and  glass  be  n  :  1 .  To  find  an  approximate  equa- 
tion between/,  _§•,  a,  fc,  and  n. 

Solution.  Since  the  angles  KGF,  HGD,  GHE,  LHT,  are 
small,  we  may  attribute  to  them  the  constant  ratio  of  incidence  and 
refraction,  which  exists  between  their  sines.  Thus  we  have, 

KGF  :  HGD  :  :  n  :  1 
LHI  :  GHE  :  :  n  :  1  ; 
consequently, 

KGF  +  LHI :  HGD  +  GHE  :  : «  :  1. 


270  Optics. 

Now  if  we  designate  the  acute  angles  by  the  letters  at  their  ver- 
tices F,  E,  D,  I,  we  have 


because  the  first  is  the  exterior  angle  of  the  triangle  FGD,  and  the 
second  of  the  triangle  EHI.  From  this  we  have 

KGF  +  LHI  =  F+D  +  E  +  I. 
We  shall  have  by  the  same  principle 

HGD  +  GHE  =  GME  =  E  -f  D; 
so  that  from  these  values  the  preceding  proportion  becomes 

F+  D  -f  E  +  I:E  -f  D  :  :n  :  1; 
or,  by  composition, 

F  +  1:E  +  D  ::n—  1:1; 
which  gives  the  equation 

(»  —  I)  E  +  (n  —  1)  D  =  F  +  1. 

Now,  on  account  of  the  smallness  of  all  these  angles,  we  shall  have, 
without  sensible  error, 

EH       EH 

E  proportional  to  -==  =  —  , 

AG__AG 

'    AD  ~  f  * 

V  AG^_AG 

'  AF~    a  ' 

,  EH  _  EH 

'  El"    a  ' 

Substituting  these  values  in  the  preceding  equation, 

(n  —  1)  EH       (n-l)AG  __  AG       EH 
'    g  f 

But,  on  account  of  the  small  thickness  of  the  glass,  the  points  G  and 
H  are  nearly  coincident  ;  and  as,  besides,  the  lines  FG,  GH,  HI, 
make  small  angles  with  the  axis,  it  follows  that  AG  •=.  BH  nearly  j 
we  may,  therefore,  divide  the  whole  equation  by  AG  or  BH,  and 
then  we  shall  have 


Mathematical  Additions.  27  JL 


which  is  the  approximate  formula  sought. 

570.  Additions  .  (1.)  Each  of  the  5  quantities,  n,f,  g,  a,  «,  may 
be  considered  as  unknown,  the  other  being  given.  This  gives  rise  to 
5  propositions  capable  of  many  important  applications. 

(2.)  The  formula  is  applicable  to  all  the  positions  of  the  points  F 
and  1  upon  the  axis,  provided  that  in  each  case,  regard  be  paid  to  the 
position  of  the  part  which  is  considered  as  given.  According  to  the 
form  and  thickness  of  the  glass,  this  position  may  be  the  same  as  in  the 
figure,  or  it  may  be  the  opposite.  If  the  glass,  for  example,  were  dou- 
ble concave,  we  should  have  to  make  /and  g  negative.  If  the  anterior 
surface  were  plane,  and  the  posterior  surface  concave,  we  should  have 

n  —  1 
to  make/  =.  oo,  which  would  give  —  -  —  =  0  ;  and  g  would  be  neg- 

ative. If  the  ray  FG  did  not  come  from  a  point  of  the  axis  before 
the  glass,  but  on  the  contrary  were  directed  towards  a  point  of  the 
axis,  behind  the  glass,  we  should  have  «  negative.  If  the  ray  FG 

were  parallel  to  the  axis,  we  should  have  a  =  00,  and  -  =  0. 

The  consequences,  therefore,  deduced  from  the  formula,  may 
serve  for  all  spherical  glasses  and  for  all  cases  in  which  they  can  be 
used. 

(3.)  Since  AG  and  BH  are  eliminated  from  the  equation,  this  is 
a  proof  that  all  rays  coming  from  the  point  .F,  and  falling  on  the 
glass,  will  unite  in  the  same  point  /,  and  consequently  produce  there 
an  image  of  F.  But  in  certain  cases,  this  image  of  F  may  be  before 
the  glass,  which  happens  when  the  formula  gives  a  negative  value 

for  a  ;  it  may  be  at  an  infinite  distance,  if  -  =  0,  that  is,  «  =  oo. 

(4.)  Putting  together  the  consequences  above  obtained,  we  find, 
that  in  each  kind  of  spherical  glasses,  a  radiant  point,  placed  in  the 
axis,  always  produces  by  refraction  an  image  situated  in  this  same 
axis,  but  sometimes  before  the  glass,  sometimes  behind  it,  and  some- 
times at  an  infinite  distance  ;  all  which  was  taken  for  granted  in  our 
method  of  construction. 

(5.)  Let  «  be  the  quantity  sought,  and  a  =  oo,  that  is,  suppose 

the  incident  ray  parallel  to  the  axis  ;  we  shall  have  -  —  0  ;  conse- 
quently, 


272  Optics. 

1        n  —  i        n—  1 

«  =  "  +  "- 

By  means  of  this  formula,  we  determine  the  distance  where  the 
rays  parallel  to  the  axis,  cut  each  other  after  refraction. 

This  distance  is  called  the  focal  distance  of  the  glass,  and  if  we 
represent  it  by  p,  we  shall  have, 

1  _  n  —  1        n  —  1 

p~  ~     ~T'' 

which  is  one  of  the  principle  formulas  of  dioptrics.  By  means  of 
this  formula  we  can  find,  in  each  case,  the  focal  distance  of  a  glass, 
by  knowing  the  radius  of  its  surface,  and  the  ratio  of  refraction. 
Many  other  useful  questions  may  be  resolved  in  the  same  manner, 
since  if  of  4  quantities  p,  n,/,  g,  three  are  given,  the  fourth  may  be 
easily  obtained  from  the  formula. 

If  the  glass  is  symmetrically  double  convex,  f—g;  consequently, 


_ 

~     - 


P  - 

Let  the  ratio  of  refraction  be  17  :  11.     Then  we  have 

.  =  *,  —  =£,»(.-.>  =  {!. 

consequently,  p  =  }|/;  that  is,  p  =f,  wanting  T^. 

If  the  glass  is  double  concave,  and  symmetrical,  we  have,  in  tin; 
same  manner,  p  =  —  }^/. 

If  the  glass  is  plano-convex,  one  of  the  radii  g,  for  example,  be- 
comes infinite  ;  consequently, 

n—  1  ,  1        n  —  1  / 

--  =  0,  and  -  =  —  -  —  or  p  =  —  -  —  . 
g     ,  P  J  «  —  1 

If  now  we  make  as  above,  n  —  1  =:  T6T,  we  have  p  =  y/,  or 
nearly  2/. 

When  we  understand  how  to  employ  algebraic  formulas,  we  easily 
see  how  the  calculation  is  to  be  made  in  all  other  cases.  The  fol- 
lowing example  may  serve  to  indicate  the  most  convenient  form  of 
calculation.  Let  us  suppose  that  the  glass  is  a  meniscus,  and  that 
its  anterior  surface  is  concave.  In  this  case  let 

/  =  —10;£=  -hi;  n  —  1  =  7«Tj 
we  shall  have 

=  ~  T!T  +  ff  =  If*  -  T?»  =  +  if*  ; 


Mathematical  Additions. 


273 


consequently,^  =  +  iif-     Performing  the  division,  we  have 

p  -  4.  o,374. 
(6.)  Since  in  general,  according  to  what  precedes, 

~f~~       ~g~  ~  a       «' 

the  first  member  of  this  equation  being  equal  to  -,  the  second  mem- 
ber also  must  be  equal  to  -  ;  and  we  shall  have  generally 


which  is  the  formula  obtained  in  catoptrics,  for  comparing  the 
focal  distance  of  a  mirror,  with  the  two  distances  of  meeting  of  the 
rays.  And  in  this  case,  as  before,  the  formula  is  applicable  to 
almost  all  optical  calculations. 

57 1 .  It  still  remains  for  us  to  consider  the  refraction  of  rays  com- 
ing from  a  point  placed  without  the  axis.     It  may  be  proved  as  well 
by  theory  as  experiment,  that  the  radiant  points  which  are  near  the 
axis,  give  distinct  images.     This  is  a  sufficient  reason  for  directing 
our  attention  to  them. 

The  line  JIB  (Jig.  87)  represents  the  profile  of  any  glass  what- 
ever, whose  optic  centre  is  C,  and  whose  axis  is  DE.  F  is  a  ra- 
diant point,  from  which  one  of  the  rays  FG,  is  refracted  in  the 
direction  GH.  If  from  the  point  F,  we  draw  the  line  FCH  through 
the  optic  centre  C,  the  refracted  ray  will  cut  it  in  some  point,  which 
we  shall  call  H.  And  consequently  the  distances  CF  and  CH  are 
connected  together  by  a  certain  law. 

572.  Theorem.     If  we  call  p  the  focal  distance,  and  if  the  radi- 
ant point  is  very  near  the  axis,  we  shall  have  (570) 

1  _      1,1 
p  ~  CF  "*"  CH' 

Demonstration.  Produce  HG  towards  /,  and  GFto  the  axis  in 
D.  If  we  designate  the  acute  angles  by  D,  K,  F,  H,  we  shall  have 
F  4-  H  =  D  4-  K,  because  each  of  these  two  sums  is  equal  to 
IGF.  But  as  F  is  near  the  axis,  the  angle  GFC  is  very  small, 
and  GC  is  nearly  perpendicular  to  FC ;  we  may,  therefore,  coa- 
sider,  without  any  material  error, 

Ekm.  35 


Optics. 

CG 

F  proportional  to  -^, 

CG 
"CH9 


CG 


If  we  substitute  these  values  in  the  equation  F  -f-  H  =  D  -{-  /£ 
we  shall  have 

^CG        CG  _  CG      CG 
PC  ^  CH~  CD+  ~CK  ' 

or,  dividing  the  whole  by  CG, 

_L  +  _L  =  JL  +  _L 

f  C  ^  Ctf        CD  ^  CK' 

Suppose  that  the  ray  does  not  come  from  F,  but  from  D,  which 
cannot  make  any  change  in  the  refracted  ray  ;  we  have,  according 
to  article  571,  (5,) 

1  _.  J_          1 

p  ~  DC  "*"  CK' 
and  consequently, 

I-    J_4-_L. 
j>  ~  FC  "*"  CH' 

which  is  tlie  formula  sought. 

573.  Addition.  It  is  obvious  from  this  formula,  that  the  results 
obtained  for  a  radiant  point  within  the  axis,  may  also  be  applied  to 
points  without  the  axis.  Hence,  1  .  Each  radiant  point  F,  placed 
without  the  axis,  produces  after  refraction  an  image  H  always  situ- 
ated in  a  straight  line  drawn  from  the  radiant  point  to  the  optic  cen- 
tre. 2.  If  through  the  points  F  and  H  we  draw  the  lines  FL  and 
HM  perpendicular  to  the  axis,  since  F  and  M  are  very  near  the 
axis,  we  may  suppose,  without  sensible  error,  that  CF  =  CL,  and 
CH  =  CM  ;  so  that  the  formula  will  be  changed  to 

1=    -1    4-J- 
p        CL  ^  CM 

But  it  hence  follows  that  if  FL  is  a  radiant  object,  eacli  of  its 
points  will  have  its  image  in  HM,  and  that  the  image  of  each  of 


Compound  Optical  Instruments.  275 

them  will  be  found  exactly  in  the  straight  line  drawn  from  this  point 
to  the  centre. 

This  is  the  second  supposition  made  in  our  method  of  construc- 
tion. And  thus  all  which  was  then  admitted  without  proof,  is  now 
rigorously  demonstrated. 


CHAPTER  XLIII. 

Compound  Optical  Instruments^ 

A.  Refracting  Telescope. 

574.  BY  means  of  spherical  glasses,  we  can  form  a  variety  of 
different  combinations,  which  make  objects  appear  larger  and  nearer 
than  they  really  are.   Such  a  combination  constitutes  what  is  called  a 
telescope.    When  it  is  composed  of  glasses  alone,  it  is  called  a  refract- 
ing telescope  ;  when  spherical  mirrors  are  used,  a  reflecting  telescope. 
The  glass  or  mirror  which  immediately  receives  the  light  of  the 
object,  is  called  object  glass,  or  object  mirror.    The  others  are  called 
eye  glasses,  and  are  designated  by  first,  second,  &c.,  beginning  with 
that  nearest  the  object  and  reckoning  towards  the  eye. 

575.  To  render  the  effect  of  a  telescope  as  perfect  as  possible, 
each  glass  must  be  exactly  centred  ;  the  axes  of  all  the  glasses  must 
be  in  the  same  straight  line ;  each  glass  must  have  its  focal  distance 
exactly  determined  by  fixed  rules,  and  especially  an  exactly  pro- 
portionate aperture.     Between  the  glasses   we  place  diaphragms, 
which  are  opaque  circles  having  a  hole  at  the  centre,  and  of  which  it 
is  very  important  to  determine  the  position  and  diameter.    Lastly,  all 
the  glasses  must  be  placed  at  distances  prescribed  before  hand ;  and 
even  the  eye  must  have  its  place  exactly  determined.     Sometimes 
the  last  eye-glass  only  is  moveable ;  but  more  frequently  all  the  eye- 
glasses are  enclosed  in  a  tube,  for  the  purpose  of  varying  their  dis- 
tance from  the  object-glass  so  as  to  suit  the  eye. 

576.  With  telescopes  we  see  distant  objects  under  a  much  greater 
angle  than  with  the  naked  eye.     The  number  which  expresses  how 
raapy  times  this  angle  is  augmented  is  called  the  magnifying  power. 

The    space   which   we  perceive   through   the   whole   system  of 
gasses  is  circular,  aud  is  called  the  field  of  view.     The  measure  of 


276  Optics. 

this  field  is  the  angle  under  which  the  eye  would  see,  without  the 
telescope,  the  whole  space  which  it  embraces  by  means  of  the  tele- 
scope. 

These  several  particulars  are  susceptible  of  mathematical  deter- 
mination ;  but  all  that  can  be  expected  in  an  elementary  work,  is  to 
deduce  and  explain  the  effects  of  telescopes  from  the  properties  of 
spherical  glasses  and  mirrors. 

577.  To  instruments  composed  of  several  glasses,  the  following 
general  remarks  are  applicable.  We  have  shown  in  the  preceding 
chapters,  that  each  spherical  glass  produces  an  image  of  the  object 
whose  rays  fall  upon  it,  but  that  this  image  may  be  sometimes  before 
the  glass,  sometimes  behind  it,  and  sometimes  at  an  infinite  distance. 
If  now  we  place  a  second  glass  behind  the  first,  so  that  their  axes 
shall  correspond,  the  image  produced  by  the  first,  will  serve  as  an  ob- 
ject to  the  second  ;  but  the  image  produced  by  the  second,  may  be 
before,  behind,  or  at  an  infinite  distance.  This  image  will  serve  as 
an  object  for  the  third  glass,  and  so  on.  Hence  we  see  that  whatever 
be  the  number  of  glasses  arranged  upon  a  common  axis,  and  what- 
ever be  their  distance,  each  glass  will  produce  a  particular  image  of 
the  object. 

Some  of  these  images  are  actually  formed,  because  the  rays 
which  belong  to  a  determinate  point  of  the  object,  actually  unite  in 
the  same  point  after  refraction.  Such  are  the  images  produced  by  a 
convex  glass  in  the  camera  obscura.  These  are  called  real  or  physi- 
cal images.  Others  do  not  actually  exist,  either  because  the  light  is 
only  propagated  as  if  it  came  from  such  an  image,  as  is  the  case  in. 
opera-glasses,  and  in  magnifiers  ;  or  because  the  rays  which  would 
produce  an  image,  are  received  by  a  new  glass,  before  the  image  has 
been  completed.  These  are  called  geometric  images.  But  cases 
are  presented  in  compound  instruments,  which  have  not  been  con- 
sidered in  the  preceding  chapter,  where  the  object  was  always  effec- 
tive. 

A  real  object,  for  example,  is  always  before  the  glass.  An  image 
which  serves  as  an  object  to  the  following  glass,  may  be  behind  this 
glass.  This  case  would  happen,  for  example,  if  instead  of  letting 
the  image  fg  be  actually  formed  (jig.  84),  we  should  receive  the 
light  upon  another  glass  placed  somewhere  between  AB  and  fg. 
This  case,  therefore,  gives  rise  to  a  particular  series  of  phenomena ; 
but  they  may  be  explained  by  the  same  method  of  construction 
which  was  employed  in  the  preceding  chapter. 


Telescopes.  277 

Phenomena  produced  by  means  of  Converging  Glasses   token  the 
Object  is  behind  the  Glass. 

578.  When  the  object  is  behind  a  converging  glass,  there  is 
always  produced  a  small  real  image  placed  very  near  the  glass. 

Let  FG  (Jig.  88)  be  the  image  which  would  be  produced  by  a 
glass  placed  in  D,  if  the  light  were  not  collected  by  the  glass  JlB, 
before  this  image  could  be  formed.  We  know  that  this  image  is 
produced  by  the  convergent  rays  which  unite  to  form  the  point  F, 
for  example.  Among  these  convergent  rays,  there  may  be  one  LC, 
which  passes  through  the  optic  centre  C,  and  which  consequently 
continues,  without  being  refracted,  in  the  direction  CF.  There 
may  also  be  one  of  these  rays  KA,  which  is  parallel  to  the  axis. 
This  is  refracted  towards  the  principal  focus  E ;  the  rays  CF  and 
AE  cut  each  other  in  the  point/;  and  there  all  the  rays  must  meet, 
which  would  have  met  at  F,  without  the  interposition  of  the  glass 
JIB ;  that  is,  if  /  is  an  image  of  JP.  Accordingly,  if  we  draw  the 
line  fgh  perpendicular  to  the  axis,  we  find  that  fg  is  the  image  of 
FG.  Meantime,  we  see  that  the  two  rays  KJl  and  LC  need  not 
actually  exist.  The  rays  which  must  represent  a  point  F,  are 
always  comprehended  in  a  small  angle ;  and  it  may  happen  that 
within  this  angle,  there  is  no  ray  parallel  to  the  axis,  and  none  which 
passes  through  the  optic  centre.  But  since  all  the  rays  which  go 
towards  F,  have  one  and  the  same  point  of  coincidence  in  /  it  is 
indifferent  whether  the  rays  made  use  of  to  find  the  situation  of  /, 
exist  or  not. 

We  see  by  the  figure  that  the  phenomenon  continues  always  the 
same  in  its  essential  parts.  In  whatever  place  the  object  FG  is  be- 
hind the  glass,  there  is  always  formed  between  the  glass  and  the 
principal  focus,  a  reduced  image,  which  is  erect  or  inverted,  accord- 
ing as  the  object  FG  is  inverted  or  erect.  Only  the  magnitude  and 
distance  of  this  image  change  with  the  distance  of  the  object. 

Upon  this  is  founded  the  effects  of  a  collector  placed  behind  a 
burning  glass  which  were  described,  article  553. 


278  Optics. 

Phenomena  produced  by  means  of  Diverging  Glasses,   when  the 
Object  rs  behind  the  Glass. 

579.  The  phenomena  produced  under  these  circumstances  are 
very  various  ;  yet  they  may  be  easily  conceived  from  what  has  been 
said  above. 

(1.)  Figure  89  represents  the  case  where  the  image  FG,  which 
takes  the  place  of  the  object,  is  within  the  posterior  focal  distance 
CE  ;  the  two  rays  KA  and  LC,  which  would  cut  each  other  in  F, 
were  it  not  for  the  glass  JIB,  after  their  passage  take  the  converg- 
ent directions  Af  and  Cf;  accordingly  a  larger  and  more  distant 
image  fg,  is  produced  behind  the  glass,  which  has  a  position  similar 
to  FG. 

(2.)  Figure  90  represents  the  case  where  the  image  FG  is  in  the 
posterior  focus  E.  Here  the  two  rays  KA  and  LC,  which  in  the 
glnss,  would  cut  each  other  in  jP,  have,  after  passing  the  glass,  the 
parallel  directions  AM  and  CF,  so  that  an  image  is  nowhere  form- 
ed ;  or,  we  may  say,  it  is  formed  at  an  infinite  distance. 

(3.)  Figure  91  represents  the  case  where  the  image  FG  is 
placed  without  the  posterior  focal  distance  CE.  In  this  case,  the 
two  rays  KA  and  LC,  after  their  passage,  have  the  divergent  direc- 
tions AM,  (  F;  if  the  lines  AM,  CF,  are  produced  before  the 
glass  sufficiently  far,  they  will  cut  each  other  below  the  axis  in/. 

The  rays  continue,  therefore,  after  their  passage  through  the  glass, 
as  if  they  came  from  an  image  fg,  placed  without  the  anterior  focal 
distance,  and  having  an  inverted  situation  with  respect  to  FG. 
But  the  magnitude  and  distance  of  this  image  may  be  very  different, 
according  as  FG  is  more  or  less  distant  from  E.  If  FG  is  very 
near  -E,  fg  is  very  distant  and  large.  If  EH  =  CE,  the  two  im- 
ages are  equally  large  and  equally  distant ;  but  if  FG  is  more  dis- 
tant,/^ is  smaller,  and  nearer  the  focus  D. 

B.  The  most  Important  Kinds  of  Refracting  Telescopes. 

580.  The  first  instrument  of  this  kind  was  invented  twice  near  the 
beginning  of  the  seventeenth  century.  Accident  led  Jansen,  a  specta- 
cle-maker  of  Middleburg,  to  the  invention  ;   and  Galileo,  who  had 
heard  of  this  invention,  succeeded,  by  the  force  of  his  own  genius, 
and  a  profound  knowledge  of  the  theory,  in  constructing  similar  in- 


Telescopes.  279 

siruments.  Hence  we  call  this  instrument  the  Holland  telescope,  or 
Galileo's  telescope.  The  object-glass,  is  a  converging  lens,  and  the 
eye-glass  is  a  diverging  lens  of  a  very  short  focal  distance.  The 
latter  is  disposed  in  such  a  manner,  that  the  inverted  image  of  distant 
objects,  produced  by  the  object-glass,  does  not  quite  reach  the 
posterior  focus  of  the  eye-glass,  which  refers  itself  to  the  case 
represented  in  figure  91.  An  eye  placed  very  close  behind  AB> 
will  see,  instead  of  FG,  the  image  fg;  but  as  in  the  telescope, 
FG  is  inverted,  fg  appears  erect.  The  telescope  magnifies  the 
apparent  diameter  as  many  times  as  the  focal  distance  of  the  eye- 
glass is  contained  in  the  focal  distance  of  the  object-glass.  Jt  can- 
not be  used  for  very  great  magnifying  powers,  because  the  field  of 
view  is  too  small ;  accordingly  it  is  now  employed  only  as  a  pocket 
telescope  or  opera-glass. 

581.  In  Kepler's  telescope,  the  inverted  image  produced  by  the 
object-glass,  is  seen  through  a  convex  glass  of  a  very  short  focal  dis- 
tance, just  as  we  look  at  a  real  object  through  a  magnifier.     As  this 
last  glass  does  not  invert  objects,  it  follows  that  with  such  a  telescope, 
which  is  the  best  yet  known,  we  see  the  objects  inverted  ;  but  in 
astronomical  observations  this  is   unimportant.      We  ascertain  the 
magnifying  power  in  the  same  manner  as  in  the  Galilean  telescope. 
If  we  would  obtain  a  great  magnifying  power  it  is  necessary  to  give 
the  instrument  an  inconvenient  length. 

582.  The  field  of  view  of  a  telescope  will  be  much  enlarged,  if  in- 
stead of  allowing  the  image,  produced  by  the  object-glass,  to  be  actu- 
ally formed,  we  receive  the  light  upon  a  pretty  large  glass  collector. 
Then  a  small  image  is  produced  behind  the  glass,  which  is  seen 
through  the  last  eye-glass,  as  through  a  single  lens.  By  this  arrange- 
ment we  lose  nothing  as  to  the  magnifying  power,  for  the  image  is 
magnified  as  much  more,  as  it  is  rendered  less,  by  the  interposition 
of  the  glass  collector. 

Instruments  constructed  in  this  manner,  which  magnify  little,  but 
embrace  a  wide  field  of  view  and  much  light,  are  called  night  glasses. 

583.  It  was  at  the  beginning  of  the  seventeenth  century,  that  a  Jesuit, 
named  Rheita,  invented  the  terrestrial  telescope,  in  which  there  are, 
beside  the  object-glass,  three  converging  glasses,  the  focal  distances  of 
which  are  short  but  equal.     The  three  eye-glasses  are  generally  en- 
closed in  one  tube ;  so  that  the  porterior  focus  of  each  of  them  shall 
coincide  exactly  with  the  anterior  focus  of  the  following  one.    When 
we  wish  to  use  this  instrument,  it  is  necessary  to  push  the  tube,  con- 


280  Optics. 

taining  the  eye-glasses,  into  the  other  tube,  so  far  that  the  image  pro- 
duced by  the  object-glass,  may  be  a  little  within  the  anterior  focal 
distance  of  the  first  eye-glass,  that  is,  the  one  farthest  from  the  eye.  In 
this  position,  an  eye  situated  behind  the  first  eye-glass,  would  see  the 
image  of  the  object  considerably  distant,  but  magnified  and  inverted. 
This  inverted  image  is  therefore  considerably  in  advance  of  the  focal 
distance  of  the  second  eye-glass,  and  consequently  it  produces  behind 
the  posterior  focus  of  this  glass,  an  erect  image  of  the  object.  Lastly, 
since  this  image  is  within  the  anterior  focal  distance  of  the  third  eye- 
glass, it  is  seen  through  this  as  through  a  simple  lens.  The  magni- 
fying power,  in  this  case,  is  measured  as  in  the  two  preceding,  by 
dividing  the  focal  distance  of  the  object  glass,  by  the  focal  distance 
of  one  of  the  eye-glasses.  To  obtain  a  great  magnifying  power  this 
instrument  also  must  have  an  inconvenient  length. 

584.  We  may  obtain  a  wider  field,  without  impairing  the  distinct- 
ness and  magnifying  power,  if  we  add  a  fourth  eye-glass,  and  em- 
ploy eye-glasses  of  unequal   focal  distances.     But  this  construction 
has  lost  much  of  its  importance  since  the  invention  of  achromatic 
telescopes. 

585.  There  are  several  kinds  of  telescopes  which  were  invented 
in  the  seventeenth  century.     In  the  45th  chapter,  we  shall  describe 
the  reflecting  telescope  of  Newton,  and  the  various  achromatic  tele- 
scopes. 

586.  When  the  focal  distances  of  the  glasses  of  a  telescope  are 
determined,  together  with  their  positions  and  apertures,  the  magnify- 
ing power,  field  of  view,  and  degrees  of  distinctness,  may  be  calcu- 
lated.     The   mathematical  process  cannot  here  be  made  known. 
But  as  it  is  important  to  know  the  magnifying  power  and  field  of 
view  of  a  telescope,  we  shall  briefly  describe  a  method  of  finding 
them  mechanically. 

587.  The  magnifying  power  may  be  determined  very  nearly,  by 
viewing  the  same  object  at  the  same  time,  through  the  telescope  and 
with  the  naked  eye,  and  comparing  the  apparent  magnitude  of  the 
two  images. 

It  may  be  determined  very  accurately  by  a  small  instrument  in- 
vented by  Ramsden,  the  nature  of  which  is  briefly  as  follows. 

When  we  direct  towards  the  heavens,  any  telescope  except  that 
of  Galileo,  and  hold  a  strip  of  paper  behind  the  last  eye-glass,  at  the 
point  where  the  eye  should  be  placed  we  see  a  luminous  circle  ex- 
actly defined.  The  point  most  favourable  for  the  distinctness  of  the 


Compound  Microscope.  281 

circle  is  found  by  trial.  We  measure  its  diameter  with  the  greatest 
care ;  then  we  measure  the  aperture  of  the  object-glass,  and  divide 
this  aperture  by  the  diameter  ;  and  thus  we  find  how  much  the  in- 
strument magnifies.  Instead  of  paper,  Ramsden  used  a  thin  plate  of 
horn.  He  marked  a  very  exact  scale  of  divisions  upon  it,  and 
attached  it  to  a  tube  which  could  be  joined  to  the  telescope,  and 
thus  he  measured,  in  a  most  convenient  and  exact  manner,  the  diam- 
eter of  the  luminous  circle. 

The  theory  of  this  ingenious  instrument  cannot  be  here  explained. 
We  shall  only  observe  that  the  luminous  circle  is  itself  an  image  of 
the  object-glass ;  hence  we  may  conclude  that  this  image  is  contain- 
ed in  the  diameter  of  the  object-glass,  as  many  times  as  the  telescope 
magnifies  distant  objects.* 

588.  The  field  of  view  of  a  telescope  may  be  estimated  by  com- 
paring its  diameter  with  the  apparent  diameter  of  an  object  viewed 
through  the  tube,  this  diameter  being  supposed  to  be  known  from 
other  experiments.  The  diameters  of  the  sun  and  moon  are  princi- 
pally used  for  this  purpose.  These  diameters  are  about  half  a  de- 
gree. Their  exact  values  may  be  found  in  treatises  on  astronomy. 

We  may  find  the  field  of  view  of  a  telescope  exactly,  by  directing 
it  towards  a  star  which  is  near  the  equator.  We  cause  the  star  to 
pass  across  the  middle  of  the  field,  and  observe  how  many  seconds 
elapse  during  the  passage.  Four  seconds  of  time  always  represent 
an  angle  of  a  minute. 


C.   Compound  Microscope. 

589.  The  compound  microscope  was  known  soon  after  the  inven- 
tion of  the  telescope,  but  its  inventer  is  unknown. 

With  respect  to  the  magnifying  power,  the  compound  miscroscope 
has  no  advantage  over  the  simple  microscope  ;  but  it  has  a  greater 
field,  more  light,  and  is  more  conveniently  used  for  small  objects. 

It  may  be  made  with  two,  three,  or  more  glasses.  The  object- 
glass  is  always  a  small  converging  lens,  whose  focal  distance  is 
never  more  than  half  an  inch.  Ordinarily  we  have  several  of  these 

*  M.  Arago  has  recently  invented  a  method  of  measuring  the  mag- 
nifying power,  which  is  very  ingenious,  and  which  is  founded  on 
double  refraction.  See  Cambridge  Optic?. 

Elem.  36 


28a  Optics. 

lenses,  whose  focal  distances  gradually  decrease,  because  here,  a« 
in  the  simple  microscope,  the  magnifying  power  is  always  greater  in 
proportion  as  the  focal  distance  of  the  object-lens  is  less. 

We  place  the  object  just  before  the  anterior  focus  of  the  object 
lens ;  consequently  a  magnified  and  inverted  image  of  the  object  is 
formed  at  a  considerable  distance  behind  the  lens.  'We  can  see  this 
image  through  a  convex  glass  of  one  or  two  inches,  as  through  a 
common  magnifier.  Such  is  the  construction  of  the  microscope 
with  two  glasses. 

But  the  combination  of  three  glasses  is  preferable.  Instead  of 
allowing  the  image  to  be  formed  by  the  object-lens,  as  in  the  case 
just  described,  we  receive  the  light,  before  its  formation,  upon  a  large 
glass  of  about  3  inches ;  so  that,  according  to  what  is  said,  article 
578,  a  small  image  of  the  object  is  produced  behind  this  glass, 
which  may  be  seen  through  a  second  eye-glass,  whose  focal  distance 
is  about  one  inch. 

It  is  not  expedient  to  employ  a  greater  number  of  glasses,  because 
the  light  is  enfeebled  by  them.  Moreover  great  care  must  be  used 
in  arranging  them  ;  otherwise  the  effect  is  indistinct. 

590.  In  a  compound  microscope  the  magnifying  power,  field  of 
view,  &c.,  are  calculated  in  the  same  manner  as  for  a  telescope  j 
by  the  focal  distances  of  the  glasses,  and  their  distances  from  each 
other ;  but  we  have  not  room  to  consider  this  subject. 

The  magnifying  power  may  be  estimated  by  the  following  experi- 
ment. We  put  in  the  microscope  a  small  object  of  exactly  known 
dimensions ;  we  look  into  the  instrument  with  one  eye  ;  and  with  the 
other,  towards  the  points  of  a  pair  of  compasses  held  at  the  distance  of 
distinct  vision.  We  open  the  points  of  the  compasses,  until  their 
distance  appears  to  be  equal  to  the  diameter  of  the  object  seen 
through  the  microscope.  We  measure  this  distance  upon  a  scale 
of  equal  parts,  and  divide  it  by  the  true  diameter  of  the  object.  We 
employ  also  for  the  same  purpose  the  measure  of  the  magnifying 
power  described  in  article  586,  and  in  exactly  the  same  manner  as 
for  a  telescope  ;  only,  for  the  number  that  gives  the  measure  of 
the  magnifying  power,  which  in  this  case  will  be  unity  or  a  fraction, 
must  be  multiplied  by  the  distance  of  distinct  vision,  that  is,  by  about 
8  inches,  and  divided  by  the  focal  distance  of  the  object-lens. 


The  Glass  Prim.  283 

CHAPTER  XLIV. 

Theory  of  Dioptric  Colours,  or  the  Decomposition  of  Light. 
The  Glass  Prism. 

591.  WE  now  proceed  to  a  careful  examination  of  the  phenomena 
of  the  dispersion  of  colours,  which  are  produced  in  every  case  of 
refraction. 

The  very  simple  apparatus  by  means  of  which  Newton  clearly 
demonstrated  the  laws  of  these  phenomena,  is  a  glass  prism,  of  which 
ABC  (Jig.  92)  represents  a  vertical  section.  Ordinarily  the  prisms 
which  are  used  in  these  experiments  are  symmetrical,  and  each  of 
their  three  angles  BAC,  ABC,  ACS,  contains  60  degrees.  Yet 
we  sometimes  use  irregular  ones.  Most  frequently  they  are  5  or  6 
inches  in  length,  so  that  we  can  look  through  them  with  both  eyes 
at  once.  When  the  light  passes  through  these  prisms,  each  ray  is 
refracted  twice  ;  namely,  at  the  anterior  surface  BA,  and  at  the  pos- 
terior one  CA ;  by  this  double  effect  the  refraction  and  dispersion 
of  the  colours  are  much  increased,  and  we  may  then  easily  examine 
the  refracted  light,  at  such  a  distance  as  we  please  behind  the  prism. 
The  angle  BAG,  formed  by  the  two  surfaces  BA  and  CA  of  the 
prism,  is  called  the  refracting  angle. 

592.  Suppose  that  we  hold  a  prism  of  this  kind  before  both  eyes 
in  a  horizontal  position,  so  that  one  of  the  angles,  BAG,  for  exam- 
ple, as  in  figure  92,  is  at  the  bottom.  If  we  then  view  objects  through 
one  of  the  refracting  surfaces,  CA,  for  example,  we  see  them  much 
lower  than  they  really  are,  and  all  the  objects  which  are  situated 
towards  the  sides,  change  place  still  more  sensibly,  so  that  a  horizon- 
tal line  appears  as  an  arc  concave  on  the  upper  side.     At  the  same 
time  the  borders  of  all  objects  appear  to  be  surrounded  with  the. 
colours  of  the  rainbow  ;  but  for  this  very  reason  they  are  indistinct. 

593.  The  experiments  made  with  a  prism  in  a  dark  room  are 
still  more  remarkable  and  decisive.     We  cause  a  small  cone  of  solar 
rays  DE  (jig.  92)  to  pass  through  a  small  opening  D,  made  in  a 
window  shutter,  and  fall  upon  the  face  BA  of  the  prism.    This  light 
is  refracted  twice  at  E  and  F,  and  always  upward.    After  refraction 
it  spreads  wider,  the  farther  it  extends.    If  this  light  is  received 


284  Optics. 

upon  a  white  screen  J^G,  opposite  the  opening  D,  we  observe  in 
VR,  where  the  light  strikes,  the  most  beautiful  phenomenon  which 
colours  are  capable  of  producing  ;  that  is,  an  elongated  image  VR, 
as  represented  in  figure  93.  It  is  nowhere  exactly  defined  ;  yet  the 
two  lateral  lines  JIB  and  DC  are  easily  distinguished  ;  nor  is  it  diffi- 
cult to  perceive  that  the  upper  and  lower  parts  terminate  in  a  semi- 
circle, although  their  contour,  and  especially  that  of  V  are  very 
indistinct.  The  entire  image  is  about  5  times  as  long  as  it  is  wide, 
and  each  point  of  its  height  is  marked  by  different  and  very  lively 
colours.  The  order  in  which  these  colours  are  disposed,  as  well  as 
the  space  they  generally  occupy,  are  indicated  approximately  by  the 
lines  which  cut  figure  93,  and  by  the  words  which  are  placed  against 
them.  Yet  the  determination  of  the  space  which  each  colour  fills, 
cannot  be  exact,  since  they  run  into  each  other  by  insensible  grada- 
tions ;  so  that,  in  fact,  the  whole  space  from  V  to  R  is  only  a  con- 
stant variation  of  colours,  in  which  we  can  only  distinguish  the  7 
predominant  shades  indicated  above.  The  image  must  be  received 
at  a  considerable  distance  from  the  prism,  at  least  12  feet,  since 
nearer  the  posterior  surface  of  the  prism,  the  image  is  perfectly 
white  in  the  middle,  and  only  coloured  towards  the  top  and  bottom  ; 
whereas,  the  more  the  light  is  dilated  by  distance,  the  more  dis- 
tinct are  the  colours.  This  image  of  colours  is  called  the  solar 
spectrum. 

594.  In  order  to  comprehend  easily  the  formation  of  the  solar 
spectrum,  we  must  examine  with  care  the  refraction  which  a  single 
ray  experiences  in  the  prism.  Let  BAC  (fig-  94)  be  the  refracting 
angle  of  a  prism  with  vertical  sections.  Let  the  ray  DE  fall  upon 
the  anterior  face  AB.  At  E  erect  the  incident  perpendicular  IH. 
It  is  evident,  from  what  has  been  said,  that  the  ray  is  refracted  up- 
ward in  the  glass.  Let  EF,  therefore,  be  the  refracted  ray.  At  F, 
where  it  reaches  the  posterior  surface,  erect  the  perpendicular  LK. 
It  will  be  seen  that  at  its  emergence  from  the  glass  it  is  again  refract- 
ed upward.  Let  FG,  therefore,  be  the  emergent  ray.  If  we  pro- 
duce the  incident  ray  DE  indefinitely  towards  JV,  and  the  emergent 
ray  until  it  meets  D./V  in  M,  the  acute  angle  GMN  is  the  quantity 
by  which  the  ray  DE  is  turned  from  its  primitive  direction,  by  the 
two  refractions  which  it  successively  experiences.  It  is  demon- 
strated by  the  calculus,  that  when  the  refracting  angle  of  the  prism 
is  not  very  large,  this  angle  GMN,  at  whatever  place  the  ray  DE 
falls,  has  a  nearly  constant  ratio  with  the  refracting  angle  BJ1C. 


The  Glass  Prism.  285 

For  example,  let  the  ratio  of  refraction  between  air  and  glass  be 
»  :  1.  We  have  almost  exactly  GMN  =  (n  —  1)  BAC;  and  as 
we  find  that  the  ratio  of  refraction  in  common  glass  is  about  3:2, 
we  have  n  =  |  ;  consequently,  n  —  1  =  1  ;  therefore, 


that  is,  by  the  effect  of  the  glass  prism,  the  ray  DE  is  turned  from 
its  primitive  direction  by  a  quantity  nearly  equal  to  half  the  refract- 
ing angle,  and  always  towards  the  opening  of  the  angle.* 

*  Demonstration.  The  formula  GMN  =  (n  —  I)  BAC,  would  be 
rigorously  exact,  if  the  angles  themselves,  instead  of  their  sines,  were 
as  n  :  1.  Upon  this  supposition,  and  observing  that  DEH  =  LEM, 
we  have 

LEM  :  LEF  ::»:!; 
whence  we  obtain 

MEF  :  LEF  :  :  n  —  1  :  1  ; 
also  at  the  other  surface 

GFK  =  LFM, 
and  we  have 

LFM  :  LFE  :  :  n  :  1  ; 
consequently, 

MFErLFE::  n  —  1  :  1. 
From  the  second  and  fourth  proportion  we  deduce 

MEF  +  MFE  :  LEF  +  LFE  :  :  n  —  1  :  1. 
We  have,  moreover, 

MEF  -j-  MFE  =  GMN, 
LEF  +  LFE  =  ILF, 
consequently, 

GMN  :  ILF  :  :  n  —  1  :  1  ; 

Now  ILF  is  the  supplement  of  ELF  ;  and  as  the  quadrilateral 
AELF  has  two  of  its  angles  E  and  F  right  angles,  ELF  is  the  sup- 
plement of  EAF  ;  consequently,  ILF  is  equal  to  EAF,  or  to  the 
refracting  angle  of  the  prism.  Thus  the  preceding  proportion  is 
changed  into  the  following  ; 

GMN  :  BAC  :  :  n  —  1  :  1, 
whence  we  obtain 

GM2V=(n—  l)  BAC. 


286  Optics. 

595.  We  here  remark  also,  that  the  use  of  the  prism  is  the  mosr 
convenient  mechanical  means  of  finding  exactly  the  ratio  of  refrac- 
tion n  :  I  ;  for  from  the  formula  GJMA*  =  (n  —  1)  BAC,  it  follows 
that 

GMN 


We  have  only,  therefore,  to  measure  these  two  angles,  which 
operation  is  attended  with  no  difficulty  ;  yet,  if  we  wish  to  determine 
their  ratio  with  extraordinary  precision,  we  must  employ  for  the 
value  of  n  a  more  rigorous  formula. 

596.  The  preceding  remarks  upon  the  phenomena  of  the  prism 
must  be  sufficient  to  give  a  clear  idea  of  it. 

Let  us  suppose,  then,  an  observer  having  his  eye  placed  at  G 
(Jig.  94.)  The  point  D,  from  which  the  ray  DEFG  proceeds, 
will  appear  to  him  in  the  direction  GF,  lower  than  it  is.  Yet  if 
each  ray  which  passes  through  the  prism  were  only  to  experience  a 
deviation  equal  to  £  BJ1C,  from  its  primitive  direction,  it  is  clear 
that  all  the  effect  of  the  prism  would  consist  merely  in  causing  all 
objects  to  appear  out  of  their  place,  by  a  quantity  equal  to  the  angle 
imlicnted.  But  our  theorem  is  only  applicable  to  the  rays  which  pass 
in  a  plane  perpendicular  to  the  prism.  Those  rays,  on  the  contrary, 
which,  when  we  look  through  the  prism  in  the  manner  directed  in  arti- 
cle 590,  come  from  the  objects  placed  at  the  sides,  are  more  strongly 
refracted,  since  they  describe  a  greater  space  in  the  prism,  and  since, 
by  meeting  the  surface  obliquely,  they  must  make,  in  the  interior,  a 
greater  angle  of  refraction.  Thus  when  we  view,  through  a  prism, 
a  straight  horizontal  line,  parallej  to  the  edge  of  the  prism,  it  must 
appear  arched  and  with  its  extremities  directed  downward,  since  the 
light  which  comes  from  these  extremities,  is  turned  more  from  its 
primitive  direction. 

After  what  has  been  said  of  the  phenomena  observed  in  a  dark 
room,  the  formation  of  the  spectrum  will  be  easily  understood. 

597.  If  we  take  away  the  prism  BAC  (Jig.  92)  and  let  the  light 
which  enters  through  the  opening  D,  fall  directly  upon  the  screen, 
we  only  see  a  round,  white,  and  ill-defined  image  of  the  sun.    Now, 
if  there  were  no  dispersion  of  colours  by  refraction,  the  whole  effect 
of  the  prism  would  consist  in  making  the  image  appear  in  VR  with 
the  same  colour,  and  with  a  very  slight  change  in  the  form  and  mag- 
nitude.    The  elongated  figure  of  the  image  VH  shows,  then,  un- 
questionably, that  the  refracted  light  FVR  has  an  ununiform  rcfrac- 


The  Glass  Prisnt.  287 

lion,  since  the  part  which  is  carried  to  V  by  refraction,  is  turned 
further  from  its  primitive  direction  than  the  part  which  arrives  at  R. 
Moreover,  since  the  image  exhibits  a  different  colour  at  each  suc- 
cessive point,  we  infer  that  the  whole  light  of  the  sun  is  divided  by 
refraction,  into  rays  of  different  colours,  and  that  the  light  of  each  of 
the  colours  has  a  ratio  of  refraction  peculiar  to  itself. 

In  the  glass  of  which  Newton's  prism  was  made,  the  ratio  of  re- 
fraction, 

For  violet  light  was  .  .  1,56  :  1 
For  the  intermediate  green  1,55  :  1 
For  the  extreme  red  .  .  1,54  :  1.* 

598.  If  all  the  light  which  passes  through  the  prism  were  violet, 
and  of  an  equal  refrangibility,  we  should  see  in  J^only  a  round  and 
violet  image  of  the  sun.  If  this  light  were  red,  we  should  see  in  R 
a  red  image,  and  so  of  the  rest.  Hence  we  conclude  that  the  elon- 
gated image  VR  (Jig.  93)  consists  properly  of  an  infinite  number  of 
round  solar  images  placed  one  above  the  other,  so  that  each  ol  them 
is  a  little  higher  than  that  which  precedes  it.  In  figure  95  we  have 
by  way  of  illustration,  represented  only  the  images  which  the  7  prin- 
cipal colours  produce ;  but  it  is  obvious  that  the  solar  spectrum 
(Jig.  93)  cannot  be  formed  simply  of  seven  such  circles,  but  of  aa 


*  The  green  rays  which  are  placed  in  the  middle  of  the  spectrum, 
have,  as  their  situation  shows,  an  intermediate  refrangibility  ;  and 
consequently  it  is  to  these  only  that  the  denominalion  mean  refran- 
gibility would  seem  to  belong.  Yet  it  is  common  to  apply  it  to  the 
lowest  yellow,  because  the  light  is  more  feeble  towards  V  than  it  is 
towards  R.  The  yellow  rays  are,  then,  really  the  mean  rays,  not 
with  respect  to  their  situation,  but  as  to  their  brightness.  Newton 
found  for  their  ratio  of  refraction  in  glass  17  :  11  j  or  1,5454  : 1  ; 
and  this  is  the  ratio  of  refraction  usually  employed  in  questions  where 
no  regard  is  paid  to  the  dispersion  of  light. 

Indeed  we  employ  the  ratio  given  by  yellow  light,  in  estimating 
refracting  powers,  without  any  error;  for  if  we  use  prisms  of  a  small 
refracting  angle,  and  which  do  not  consequently  produce  a  sensible 
dispersion,  we  find  precisely  the  same  result,  as  I  have  proved  by 
experiment. 


288  Optics. 

infinite  number,  since  otherwise,  the  lateral  lines  AB  and    CD, 
would  not  appear  straight.* 

599.  These  principles  sre  confirmed  by  a  multitude  of  experi- 
ments. If  at  some  distance  from  the  prism,  we  receive  the  coloured 
light  with  a  pretty  large  converging  glass,  we  find  white  light  at  the 
focus ;  beyond  the  focus  the  different  colours  appear  in  an  inverted 
order.  If  we  place  a  second  prism  near  the  first,  but  in  a  contrary 
situation,  all  the  light  is  again  refracted  downward  in  the  same  ratio 
as  it  was  refracted  upward  by  the  first,  and  the  light  emerges  white 
from  the  second  prism.  We  can  examine  each  of  the  colours  sepa- 
rately, by  placing  at  some  distance  behind  the  prism  a  black  sur- 
face, having  a  narrow  horizontal  aperture,  through  which  only  a  thin 
uniform  section  of  the  solar  spectrum  can  pass.  Each  colour  may 
be  separated  still  better  by  placing  a  second  surface  perforated  hori- 
zontally at  some  distance  behind  the  first,  so  as  to  transmit  a  still 
narrower  line  of  light  than  the  first.  In  this  way  we  are  enabled  to 
examine  any  colour  we  please.  Having  separated  it  from  the  rest 
we  can  cause  it  again  to  pass  through  a  prism  and  determine  its 
ratio  of  refraction.  We  are  thus  able  to  separate  two  or  more  col- 
ours from  the  rest,  and  afterwards  to  unite  them  with  a  converging 
glass,  or  metallic  mirror,  &c. 

GOO.  The  following  question  has  been  much  agitated.  Into  how 
many  colours  is  the  white  light  of  the  sun  divided  by  the  prism  ? 
Newton  in  his  Optics,  has  affirmed  that  there  are  innumerable  shades 
from  the  darkest  violet  to  the  brightest  red,  and  that  each  of  these, 
however  feeble,  has  a  particular  ratio  of  refraction.  This  is  the  only 
tenable  opinion. 

601.  The  observations  of  Newton  prove  that  the  bodies  which 
appear  white,  reflect  equally  all  the  colours  of  the  spectrum,  while 
those  which  appear  coloured,  are  so,  because  they  reflect  certain  rays 
and  absorb  the  rest.  Bodies  which  appear  black,  are  those  which 
absorb  nearly  all  the  light  which  they  receive.  These  remarks  are 
not  hypothetical,  but  rest  upon  well  established  facts. 

*  The  straight  line  is  the  common  tangent  of  all  the  circles,  be- 
cause they  have  an  equal  diameter,  and  it  is  produced  by  their  con- 
tinual intersections. 


Colours  produced  by  Thin  Lamina.  £39 


Colours  produced  by  Thin  Lamina. 

602.  The  prism  is  not  the  only  means  of  decomposing  the  solar 
light  into  different  colours.     Thin  laminae  of  transparent  bodies,  as 
soap  bubbles,  produce  similar  effects.*   Newton  made  many  experi- 
ments upon  th;s  subject,  and  ascertained  that  the  effect  in  question 
is  common  to  thin  laminae  of  all  transparent  bodies,  not  except- 
ing air.     He  showed  that  under  all  circumstances  the  colour  has  a 
constant  relation  to  the  thickness  of  the  lamina ;  and  that,  conse- 
quently, for  each  different  thickness  there  is  a  determinate  order  in 
the  situation  of  the  colours.  But  neither  he,  nor  any  of  those  who  have 
followed  him,  has  been  able  to  reduce  the  formation  of  these  colours 
to  a  theory  so  simple  as  that  of  the  formation  of  the  colours  of  the 
prism.     Consequently,  we  must  be  contented,  as  to  this  phenome- 
non, with  the  following  general  consideration  ;  namely,  that  there 
must  be,  in  a  thin,  transparent  lamina,  very  various  refractions  and 
reflections ;  and  this  enables  us  to  understand  why  we  perceive  a 
decomposition  of  colours. 

603.  Newton  deduced  from  these  observations  an  hypothesis  re- 
specting colours.     He  supposed  each  body  to  be  composed  of  very 
thin  transparent  laminae,  and  each  of  these  laminae  to  give  a  particu- 
lar colour  suited  to  its  thickness.     And  it  is  true  that  there  are  phe- 
nomena which  cannot  be  otherwise  explained.     The  changing  col- 
ours of  mother  of  pearl,  Labrador  stone,  &r.,  must  be  referred  to 
such  a  cause.     Glass  and  all  other  transparent  colourless  bodies, 
appear  white  when  they  are  very  finely  pulverised  because  each 
of  the  particles  sends  off  decomposed  light,  and  white  light  is  pro- 
duced  by  their  mixture.     Yet,  there  are  many  obstacles  when  we 
undertake  to  generalize  the  applications  of  this  hypothesis.    Accord- 
ing to  all  appearances  there  is  a  kind  of  chemical  affinity,  by  means 
of  which  each  body  attracts  certain  constituent  principles  of  light,  and 


*  Some  explanation  is  here  necessary.  It  is  true,  that  thin  laminae 
decompose  the  light,  but  not  into  its  simple  rays.  The  prism  alone 
has  this  power.  The  colours  reflected  by  thin  laminae  are  compound, 
and  may  be  divided  by  the  prism.  Some  remarks  on  Newton's 
theory  of  colours  may  be  found  at  the  end  of  the  chapter. 

Elem.  37 


290  Optics. 

combines  them  with  itself,  so  that  only  the  others  can  be  reflected 
in  conformity  to  the  mechanical  laws  of  optics.* 


General  Observations  upon  Aetofon'j  Theory  of  Colours. 

604.  The  essential  part  of  this  theory  consists  in  incontestiblc 
facts  and  in  the  consequences  which  are  naturally  deduced  from 
them.  Accordingly,  what  is  important  in  it  remains  invariable.  But 
it  has  often  been  ill  understood  and  falsely  applied. 

In  particular,  men  have  confounded  the  colours  produced  by  the 
decomposition  of  solar  light,  with  those  of  material  substances,  and 
applied  to  these  what  Newton  has  advanced  respecting  the  other. 
It  is  true,  that  the  colours  of  all  colouring  matter  are  produced  by 
the  reflection  of  differently  coloured  light ;  but  as  it  is  probable  that  no 
body,  and  consequently  no  colouring  substance,  reflects  light  of  one 
simple,  fundamental  colour,  we  cannot  expect  that  the  colours  of  these 
substances  should  observe  the  same  laws  as  those  of  the  solar  spec- 
trum. Yi-t  it  may  be  demonstrated  that  the  purest  artificial  colours 
resemble,  to  a  certain  point,  the  colours  of  solar  light.  For  this 
purpose  we  use  a  plate  containing  7  divisions,  in  which  the  colours 
of  the  solar  spectrum  are  imitated  as  nearly  as  possible.  When  we 
turn  the  plate  with  great  rapidity,  it  appears  perfectly  white.  This 
is  explained  in  the  following  manner.  The  successive  impressions 
made  upon  the  retina  continue  for  a  certain  time.  Consequently, 
we  experience  nearly  the  same  effect,  when  colours  succeed  each 
other  very  rapidly,  as  when  their  rays  come  simultaneously  to  the 
eye  and  are  actually  confounded. 

Another  remark  which  must  not  be  omitted  in  considering  the 
prismatic  colours,  is  the  following.  When  we  view  a  large  surface 
of  one  single  colour,  through  the  prism,  this  surface  appears  uniformly 
coloured  in  the  middle,  although  this  uniform  colour  is  actually  com- 
pound, according  to  the  sense  in  which  we  have  considered  that 
term  in  article  599.  This  is  easily  explained.  Each  ray  coining 
from  the  surface  is  in  fact  decomposed  by  the  prism  into  different 
colours ;  but  these  different  colours,  which  come  from  different 


*  It  is  true,  that  this  mode  of  representing  the  phenomena  seems 
more  simple  at  first  view  ;  but  when  we  examine  the  subject  more 
thoroughly,  we  find  it  iufiuitely  less  probable  than  that  of  Newton. 


Effects  of  the  Dispersion  of  Colours  in  Optical  Glasses.     291 

neighbouring  points,  again  combine  together,  being  partially  super- 
posed upon  each  other,  so  as  to  form  in  these  points  a  single  colour. 
Only  at  the  extremities  of  the  surface,  where  two  compound  colours, 
blue  and  red,  for  example,  touch  each  other,  we  see  the  colours  of 
the  rainbow ;  and  this  is  not  produced  by  the  decomposition  of  one 
of  the  colours  which  fall  upon  the  surface,  but  by  a  mixture  of  sim- 
ple colours,  arising  from  the  decomposition  of  the  two  colours  which 
touch  each  other. 

In  the  same  manner,  we  can  explain  the  formation  of  all  the  col- 
ours seen  through  the  grism. 


Effects  of  the  Dispersion  of  Colours  in  Optical  Glasses. 

605.  Let  AE  (Jig.  96)  be  a  converging  glass ;  at  a  great  distance 
before  it,  let  there  be  an  object  CD,  the  white  light  of  which  falls 
upon  the  glass  ;  in  this  case,  according  to  art.  556,  557,  an  inverted 
image  KHis  produced  at  the  posterior  focus  of  the  glass.     But  it  is 
obvious,  without  calculation,  that  the  focal  distance  of  a  glass  de- 
pends upon  the  ratio  of  refraction,  and  that  it  becomes  shorter  as  the 
refracting  power  becomes  greater.     Moreover,  we  have  seen  that 
the  violet  rays  are  refracted  more  than  the  red  ;  it  is  clear,  then, 
that  the  different  colours  which  compose  the  light,  cannot  have  the 
same  focus.     Let  V,  therefore,  be  the  focus  of  the  violet  rays,  and 
7?  that  of  the  red  rays  ;  it  is  obvious  that  the  white  light  of  the  object 
CD,  will  produce  in  V  a  violet  image  FG  of  this  object,  and  in  R 
a  red  image  L  \l  of  the  same  object ;  and  between  these  two,  images 
of  the  intermediate  colours  of  the  spectrum  ;  so  that  between  FG 
and  JLJW,  there  will  be  a  great  number  of  images  placed  one  upon 
the  other  and  variously  coloured.     Accordingly,  if  the  eye  of  the 
observer  is  placed  beyond  LM,  and  views  this  image,  he  will  not 
see  it  exactly  defined  in  any  of  its  parts ;  the  indistinctness  will  in- 
crease from  the  centre  to  the  borders ;  the  red  will  pass  beyond  all 
the  rest,  and  the  entire  image  will  appear  to  be  fringed  with  the 
hues  of  the  rainbow.     This  effect  must  take  place  in  all  the  images 
produced  by  spherical  glasses,  and  it  becomes  more  striking,  in  pro- 
portion as  the  light  is  more  refracted  ;  that  is,  according  as  the  ob- 
ject is  more  magnified. 

606.  We  know,  therefore,  now  two  causes  of  the  indistinctness 
which  takes  place  in  all  optical  instruments. 


292  Optics. 

( 1 .)  The  first  consists  in  this,  that  there  is  no  curvature  in  which 
all  the  rays  that  come  from  one  point,  are,  under  all  circumstances, 
exactly  united  again  in  a  single  point ;  and  that  especially  the  spheri- 
cal curvature  which  we  give  to  glasses,  can  never  perfectly  effect 
such  a  union  of  rays  of  the  same  nature.  This  imperfection  is  com- 
mon to  lenses  and  mirrors  ;  it  is  called  the  aberration  of  sphericity, 

(2.)  The  second  cause  of  confusion  depends  upon  the  circum- 
stance above  considered.  It  consists  in  this,  that  instead  of  a  single 
image,  there  is  an  infinite  number  of  images  differently  coloured  and 
placed  nearly  behind  one  another.  This  last  cause  of  indistinctness 
is  of  much  greater  importance  than  the  other  ;  but  it  exists  only  in 
glasses,  and  not  in  metallic  mirrors ;  it  is  called  the  aberration  of 
refrangibility. 

007.  Experience  shows,  however,  that  the  eye  is  capable  of  bear- 
ing very  great  aberrations  of  each  kind,  with  less  inconvenience  as  to 
distinctness,  than  we  should  expect  from  theoretic  principles.  Nev- 
ertheless, in  the  construction  of  optical  instruments,  we  pay  so  much 
regard  to  these  defects,  as  always  to  preserve  certain  ratios  between 
the  focal  distance  and  the  aperture  of  the  object-glass ;  nor  can  we 
neglect  this,  without  essentially  impairing  the  distinctness  of  the  im- 
age. Hence  the  making  of  a  good  compound  optical  instrument,  is 
a  difficult  work,  and  requires  a  thorough  acquaintance  with  the 
science. 


CHAPTER  XLV. 

Reflecting  Telescope  and  Achromatic  Lenses. 

608.  THE  history  of  optics  is  so  instructive  and  interesting  to 
those  who  would  observe  the  progress  of  the  human  mind,  that  in 
what  remains  of  this  treatise,  we  shall  adopt  the  historical  form,  and 
only  intersperse  such  theoretical  remarks  as  may  be  necessary  for 
the  understanding  of  the  subject. 


State  of  Optics  before  Newton's  Time. 

609    Before  the  exact  law  of  refraction  was  known,  men  wr-re 
ignorant  of  the  causes  of  indistinctness  in  optical  instruments.    When 


Errors  of  Newton  in  the  Theory  of  Colours.  293 

Snellius  had  discovered  this  law,  Descartes  remarked  the  aberration 
of  sphericity,  which  is  the  slightest  cause  of  indistinctness.  But  he 
\vas  deceived  in  supposing;  that  he  could  remedy  this  defect,  by  em- 
ploying other  than  spherical  curvatures ;  and  this  mistake  continued 
till  our  times. 


Errors  of  Newton  in  the  Theory  of  Colours. 

610.  Newton  ascertained  that  the  dispersion  of  colours  is  the 
most  important  cause  of  indistinctness  in  optical  instruments  ;  and  so 
long  as  mankind  appreciate  the  sciences,  his  researches  in  optics  will 
be  regarded  as  models  of  accuracy  and  sagacity.     Unfortunately  he 
did  not  bring  them  to  a  complete  termination,  and  by  only  bestowing 
a  passing  glance  upon  one  circumstance,  of  which  he  did  not  suspect 
the  importance,  he  fell  into  some  remarkable  errors,  which  have 
been  productive  of  numerous  evil  consequences. 

611.  Newton  made  all  his  experiments  upon  the  dispersion  of 
colours,  with  prisms  of  a  single  kind  of  glass.     To  complete  his 
researches,  he  should  also  have  observed  the  dispersion  of  colours 
through  other  transparent  media.     In  the  second  part  of  the  first 
book  of  his  Optics,  he  touched  lightly  upon  this  subject ;  but  he  was 
deceived  in  respect  to  three  circumstances. 

(1.)  He  made  an  erroneous  assertion.  He  says,  that  he  caused 
light  to  pass  through  water  and  glass,  varying  in  many  ways  the 
refracting  surface  ;  and  that  he  found  the  emergent  light  to  be 
always  coloured  when  it  was  not  parallel  to  the  incident  light ;  and 
that,  on  the  contrary,  it  was  always  uncoloured  when  restored  to 
parallelism.  The  incorrectness  of  this  remark  has  since  been  ascer- 
tained. 

(2.)  He  tacitly  supposed,  without  an  experimental  examination, 
that  the  dispersion  of  colours  is  subject  to  the  same  laws  in  all  trans- 
parent media ;  and  consequently,  he  thought,  that  since  he  had  ob- 
served with  so  much  precision,  the  dispersion  of  colours,  in  ordinary 
mirror  glass,  it  was  only  necessary  for  other  transparent  media.,  to 
examine  the  ratio  of  refraction  for  the  mean  rays,  and  then,  by  com- 
paring this  ratio  with  that  of  the  glass  which  he  had  used,  he  might 
d  duce,  by  proportion,  the  ratio  of  refraction  for  the  other  colours  of 
the  spectrum.  Subsequent  inquiry  has  proved  this  reasoning  to  be 
incorrect. 


:J'J1  Optics. 

(3.)  He  deduced  from  the  experiment  mentioned  at  the  beginning 
of  this  article,  a  law,  by  which  the  dispersion  of  colours  in  two  differ- 
ent media  might  be  compared.  He  considered  this  law  as  univer- 
sally true ;  but  it  has  since  been  ascertained  that  it  only  approaches 
the  truth,  for  very  small  angles  of  refraction  ;  and  that  even  granting 
the  experiment  were  precise,  no  principles  regulating  the  dispersion 
of  colours  could  be  deduced  from  it. 

612.  If  these  ideas  of  Newton  had  been  conformable  to  truth,  it 
would  have  followed  as  a  necessary  consequence,  that  the  effect  of 
dispersion  in  optical  instruments,  could  not  by  any  means  be  reme- 
died ;  for,  in  order  to  remedy  this  effect,  it  would  be  necessary  to 
dispose  the  instrument  in  such  a  manner,  that  each  ray  when  it 
emerged  from  the  last  glass,  should  be  parallel  to  the  direction  which 
it  had  before  it  entered  the  object  glass.  But  this  would  frustrate 
the  design  of  the  instrument,  for  we  could  see  an  object  only  as  far 
as  with  the  naked  eye,  and  with  much  less  clearness.  Newton, 
therefore,  abandoned  refracting  telescopes,  because  he  supposed 
them  incapable  of  a  high  degree  of  improvement.  But  the  errors 
of  this  great  man  have  been  attended  with  one  good  effect ;  namely, 
that  of  leading  him  to  the  invention  of  a  reflecting  telescope,  com- 
monly called  the  Newtonian  telescope. 


Reflecting  Telescope. 

613.  The  essential  part  of  Newton's  reflecting  telescope  is  a  con- 
verging metallic  mirror,  which  takes  the  place  of  an  object-glass ; 
it  is  attached  to  the  bottom  of  a  tube,  the  length  of  which  is  equal  to 
the  focal  distance  of  the  mirror,  in  such  way  that  the  polished  surface 
is  turned  towards  the  opening,  and  thus  toward  external  objects.  If 
the  common  axis  of  this  mirror  and  of  the  tube  is  directed  towards 
a  distant  object,  a  small  inverted  image  of  the  object  is  formed  at 
the  focus.  But  instead  of  allowing  this  image  to  be  actually  formed, 
we  receive  the  light  at  a  distance  from  the  focus  nearly  equal  to  the 
radius  of  the  tube,  upon  a  small  plane  mirror,  attached  by  a  thin  sup- 
port, to  the  middle  of  the  axis  of  the  tube,  and  making  with  this 
axis  an  angle  of  45  degrees.  This  small  mirror,  therefore,  reflects 
the  light  which  it  receives  from  the  large  mirror  ;  and  thus  the  imnire 
which  the  large  mirror  would  have  produced,  is  formed  by  the  small 
one  on  the  side  of  the  tube,  which  lias  an  opening  at  this  place,  so 


Reflecting  Telescope.  295 

that  we  can  view  the  image  with  a  microscopic  lens,  as  in  Kepler's 
telescope. 

As  no  dispersion  is  produced  by  the  reflection  of  metallic  mirrors, 
and  as  the  aberration  of  sphericity  is  very  small,  the  image  presented 
by  such  a  mirror  is  incomparably  more  distinct  than  the  image  pro- 
duced by  an  object-glass.  This  instrument,  therefore,  admits  of  a 
much  greater  magnifying  power,  and  experience  has  proved,  that 
with  a  reflecting  telescope  of  only  a  few  feet  in  length,  we  can  see 
as  far  as  with  a  refracting  telescope  of  a  hundred  feet  in  length. 

614.  Gregory  improved  the  Newtonian  telescope,  by  introducing 
instead  of  a  small  plane  mirror  placed  obliquely  to  the  axis,  a  small 
converging  mirror  having  its  polished  surface  turned  towards  that  of 
the  large  one.     This  small  mirror  is  so  disposed  that  its  surface  is  a 
little  without  the  focal  distance  of  the  great  mirror.     Accordingly, 
if  \ve  consider  the  inverted  image  produced  by  the  object  mirror,  as 
an    object  throwing  its  light  upon  the  small  mirror,  it  is  obvious 
(Jig.  65)  that  the  latter  will  produce  a  second  image,  a  little  larger 
and  inverted.     The  small  mirror  may  be  so  placed,  that  this  image 
would  fall  behind  the  great  mirror,  if  the  light  could  pass  through  it. 
To  effect  the  formation  of  this  image,  we  make  a  circular  aperture 
at  the  centre  of  the  great  mirror  of  nearly  the  same  dimensions  with 
the  small  mirror,  through  which  the  light,  reflected  by  it,  is  perceived. 
There  would,  therefore,  be  produced  behind  the  great  mirror,  an 
image  of  distant  objects ;  but  before  it  is  formed  we  receive  the 
light  by  means  of  a  convex  glass.     In  this  way  the  rays  are  concen- 
trated into  a  smaller  image,  which  is  seen  through  a  microscopic 
lens.     The  principal  advantage  of  this  ingenious  instrument   over 
that  of  Newton,  is,  that  it  exhibits  the  objects  erect  and  in  the  direc- 
tion in  which  they  are  actually  situated. 

615.  To  obtain  the  greatest  possible  magnifying  power,  and  to  rem- 
edy the  want  of  light  which  is  common  to  all  reflecting  telescopes,  the 
celebrated  Herschel  added  to  the  Newtonian  telescope  a  modifica- 
tion which  is  applicable  only  to  very  large  instruments.     He  entirely 
dispensed  with  the  small  mirror,  and  directed  the  tube  of  the  tele- 
scope towards  an  object  not  in  the  axis,  but  a  little  above  it.     Let 
the  object,  for  example,  be  a  star,  and  conceive  a  ray  coming  from 
this  star  to  the  centre  of  the  mirror.     In  this  case  the  tube  must  be 
so  placed,  that  this  ray  shall  pass  very  near  its  upper  edge  ;  but  then 
this  ray  must  manifestly  be  reflected  towards  the  lower  edge  of  the 
opening  in  the  tube  j  and  the  length  being  equal  to  the  focal  distance 


296  Optics. 

of  the  mirror,  an  image  of  the  star  will  be  formed  near  the  lower 
edge  of  this  opening.  Accordingly,  we  can  view  this  image  directly 
with  a  lens,  provided  the  diameter  of  the  tube  is  so  large  thai  the 
part  which  the  head  of  the  observer  occupies,  is  inconsiderable  com- 
pared with  the  whole  opening.  This  arrangement  is  favourable  to 
distinctness,  brightness,  and  magnifying  power. 


Ingenious   Researches   of  Euler  ;    his   Errors. — Dollond. — Klin- 
genstiern. 

616.  The  errors  of  Newton  passed  unobserved  for  50  years; 
and  even  the  great  Euler,  the  most  profound  analyst  of  the  last 
century,  seemed  not  to  have  known  the  experiment  of  Newton  and 
the  consequences  which  are  deduced  from  it,  when  in    1747,  ho 
came  to  the  conclusion,  from  the  mere  inspection  of  the  human  eye, 
that  it  would  be  possible  to  remedy  the  dispersion  of  colours  pro- 
duced by  refraction,  since  this  defect  does  not  exist  in  our  eyes. 
His  sagacity  led  him  to  perceive,  in  the  combination  of  many  trans- 
parent substances,  the  means  employed  by  nature  to  produce  this 
effect.     He  thought  it  possible  to  imitate  such  an  arrangement  by 
placing  one  upon  the  other  two  convex-concave  glasses,  and  filling 
the  interval  between  them  with  water.     For  this  important  object  he 
employed  all  the  resources  of  analysis  ;  but  in  order  to  succeed,  it 
would  be  necessary  that  the  force  with  which  water  disperses  col- 
ours, should  have  been  determined  with  as  much  exactness  as  that 
of  the  glass  had  been  by  the  experiments  of  Newton. 

Two  methods  suggested  themselves  to  Euler ;  experiment  and 
theoretical  considerations.  He  chose  the  latter.  He  supposed  with 
Newton,  that  the  dispersion  of  colours  is  subject  to  the  same  law  in 
all  refracting  media  ;  he  strove  to  discover  this  law ;  he  found  one 
which  satisfied  all  the  conditions  that  could  be  required,  and  proved 
it  to  be  the  only  one  that  had  this  advantage.  This  law  was  entirely 
different  from  that  of  Newton,  but  it  appears  that  Euler  had  no 
knowledge  of  the  latter.  He  calculated,  therefore,  according  to  his 
own  law,  how  the  two  faces  of  an  object-glass,  composed  of  glass  and 
water,  ought  to  be  disposed,  in  order  to  give  uncoloured  images. 

617.  The  inquiries  of  Euler  created  a   great  sensation.     The 
most  skilful  artists  attempted  to  execute  object-glasses  according  to 
his  principle,  but.  without  success.     The  elder  Dollond,  an  oxcol- 


Achromatic  Combinations.  297 

lent  English  artist,  first  perceived  the  contradiction  which  existed 
between  the  laws  of  Newton  and  Euler ;  and  as  those  of  Euler  did 
not  appear  to  be  confirmed  by  experiment,  he  thought  that  the  truth 
was  on  die  side  of  Newton.  Euler,  without  examining  the  experi- 
ments and  calculations  of  Newton,  contented  himself  with  demon- 
strating by  a  rigorous  course  of  reasoning,  that  his  law  was  the  only 
possible  one,  and  attributed  die  failure  of  practical  attempts  to  the 
great  difficulty  of  execution. 

618.  Klingenstiern,  a  Swedish  geometer,  subjected  the  assertion  of 
Newton  to  a  rigorous  examination,  and  found  thai  he  deduced  from 
Newton's  experiments  not  one  single  law,  but  a  multitude  of  contra- 
dictory Jaws.     Hence  he  concluded  that  there  must  be  some  error 
in  this  experiment. 

619.  This  induced  Dollond  to  repeat  the  experiment  of  Newton. 
He  found  it  to  be  incorrect,  but  at  the  same  time  satisfied  himself 
that  the  law  of  Euler  was  not  exact,  since  die  results  ol  his  experi- 
ment did  not  conform  to  it.     Yet  as  the  opinion  of  Euler  respecting 
the  impossibility  of  uncoloured  refraction,  deduced  from  the  structure 
of  the  eye,  appeared  plausible,  he  undertook  a  new  course  of  ex- 
periments to  verify  it.     He  found  diat  the  union  of  glass  and  water 
would   not  be  adapted  to  such  a  purpose.     He  examined  different 
kinds  of  glass,  and  found  some  which  refract  light  and  disperse  the 
colours   much  more   than    ordinary   glass.     After   repeated   trials, 
he  obtained  from  two  prisms  which  were  placed  one  against  the 
other,  with  opposite  refracting  angles,  an  emergent  ray  which  was 
uncoioured,  although  the  refraction  was  very  considerable.     One  of 
these  prisms  was  English  crown  glass,  which  is  a  kind  of  mirror 
glass  of  a  greenish  colour  ;  this  had  a  refracting  angle  of  30°.    The 
other  was  flint  glass,  a  kind  of  white  glass,  containing  much  oxyde  of 
lead.     The  refracting  angle  of  this  was  19°. 

620.  This  experiment  convinced  Dollond  of  the   possibility  of 
obtaining  an  object-glass  which  should  give  uncoloured  images,  by 
employing  these  two  kinds  of  glass.     He  effected  this  object  by 
uniting  a  convex  lens  of  crown  glass  and  a  concave  one  of  flint  glass. 
Thus  he  became  the  inventor  of  achromatic  telescopes,  or  refracting 
telescopes  that,  represent  the  object  without  colour  ;  that  is,  with  its 
natural  colour. 

621.  Euler  completely  atoned  for  his  error,  which  led  to  this 
interesting  discovery,  by  executing  a  task,  which  he  would  not  per- 
haps otherwise  have  undertaken.     He  reduced  to  general  and  very 

Ehm.  38 


298 


Optics. 


simple  formulas,  not  only  the  whole  theory  of  the  aberration  of  re- 
frangibility,  but  also  the  much  more  difficult  one  of  the  aberration  of 
sphericity.  Accordingly,  we  can  now  calculate,  without  difficulty,  the 
effect  of  these  two  causes  of  confusion,  for  each  position  of  the  glass.' 
He  showed,  moreover,  that  a  triple  object-glass,  composed  of  two 
convex  lenses  of  crown  glass,  separated  by  a  double  concave  one  of 
flint  glass,  would  have  a  great  advantage  over  that  of  Doliond.  He 
ascertained  what  would  be  the  best  arrangement  of  the  eye-glass,  if 
such  an  object-glass  were  used  ;  and  above  all,  he  gave  such  a  gen- 
erality to  all  his  researches,  that  they  may  now  be  applied  to  all  opti- 
cal instruments.  Finally,  he  showed  that  though  theory  might  some- 
times err,  yet  when  properly  directed  it  extended  much  farther  than 
the  method  of  simple  experiment. 

To  those  who  would  be  farther  acquainted  with  this  part  of  the 
subject,  we  recommend  Priestley's  History  of  Optics  ;  a  work  which 
is  at  once  interesting  to  the  profound  inquirer,  and  yet  adapted  to 
the  humblest  capacity. 

622.  Strictly  speaking,  the  error  of  Euler  consisted  in  having 
sought  a  law  where  there  is  none.     For  it  is  manifest  from  the  ex- 
amination of  several  kinds  of  glasses,  that  the  different  ratios  which 
are  found  between  the  refraction  of  light  and  the  dispersion  of  col- 
ours, do  not  depend  upon  any  general  law,  but  simply  upon  the  par- 
ticular properties  of  refracting  substances  ;  and  consequently,  that 
we  can  only  ascertain  them  in  each  particular  case,  by  direct  experi- 
ment.    Nothing  can  better  evince  the  justness  of  this  opinion,   than 
the  interesting  experiments  made  by  Professor  Zeiher  of  Peters- 
burgh,   upon  the  different  kinds  of  glass.     He  ascertained  that  an 
addition  of  oxyde  of  lead,  \\ould  considerably  affect  the  dispersive 
power,  while  tin;  mean  refraction  remained  very  nearly  the  same. 
The  contrary  effect  is  produced  by  the  addition  of  alkali. 

623.  The  construction   of  achromatic  telescopes  is  not  without 
difficulty ;    and  although  they    are  made  in    many    places  besides 
England,  yet  the  English  alone  possess  the  flint  glass  which  is  em- 
ployed in  them.     Hitherto  no  artist  has  been  able  to  make  large 
instruments  of  this  kind  ;  and  this  is  the  reason  why  Herschel,  as  we 
before  observed,  had  recourse  to  the  reflecting  telescope  to  obtain 
very   great  magnifying   powers.     For  instruments  of  the  ordinary 
dimensions,  those  which  are  made  according  to  these  principles  have 
very  decided  advantages,  not  only  over   common   refracting  tele- 


Mathematical  Additions.  299 

scopes,  but  also  over  reflecting  telescopes ;  as  to  distinctness,  degree 
of  light,  and  extent  of  the  field  of  view.* 

In  compound  microscopes  it  is  not  possible  to  make  the  object- 
lens  achromatic,  because  the  glasses  of  which  it  must  be  composed, 
would  be  so  small  that  they  could  not  be  executed  with  exactness. 


Mathematical  Additions. 

624.  We  now  proceed  to  add  some  mathematical  reasonings, 
which  will  aid  us  in  forming  precise  ideas  respecting  the  theory  of 
colours,  and  the  possibility  of  producing  achromatic  images. 

625.  Experiment.      Let   there    be    any   two   media,    ABCD, 
CDEF,  (fig.  97)  terminated  by  plane  surfaces ;  and  let  there  be 
above  AB  and  below  EF,  a  medium  whose  refracting  power  is  dif- 
ferent.    If  we  suppose  a  ray  GH,  which  is  refracted  at  the  points 
H,  /,  K,  the  emergent  ray  KL,  is  always  parallel  to  the  incident 
ray  GH,  and  consequently  uncoloured. 

626.  Theorem.     Let  n  :  1  be  the  ratio  of  refraction  between  air 
and  any  medium  A  ;  and  let  m  :  1  be  the  ratio  of  refraction  between 
air  and  another  medium  B  ;  the  ratio  of  refraction  between  A  and 

B,  will  bem-. 

Demonstration.  Suppose  that  there  is  air  above  JIB  and  below  EF 
(fig.  97) ;  between  JIB  and  CD  let  there  be  the  medium  A,  and 
between  CD  and  EF  the  medium  B ;  through  /,  If,  draw  the  per- 
pendiculars MN,  OP,  QR  ;  and  call  the  ratio  of  refraction  between 

A  and  B,  -  ;  we  have, 

y 

sin  GHM  :sin  A'fl/::n:l, 

sin  HIO     :  sin  P1K  :  :  x  :  y, 

sin  IKq    :  sin  RKL  :  :  1  :  m. 

Now  JYHI  =  HIO,  and  PIK  =  IKQ',  also,  (article  625) 
GHM  =  RKL  ;  we  have  then,  compounding  these  three  propo- 
sitions ; 


*  Since  the  publication  of  this  work  the  best  kind  of  flint  glass  has 
been  made  in  France  and  Germany. 


300  Optics. 

1  :  1  :  :  nx  :  my  ; 
consequently,  nx  —  my  ;  whence 

x  :  y  :  :  m  :  n. 

627.  Theorem.  Let  Otftf  and  .##D  (/#.  98)  be  the  perpen- 
dicular sections  of  two  prisms  of  different  refracting  powers,  placed 
one  against  the  other.  Let  the  ratio  of  refraction  of  the  first  be 
n  :  1  ;  that  of  the  second  mil.  Let  a  ray  EFGHIbe  refracted  by 
these  prisms,  as  in  the  figure.  Produce  the  incident  ray  EF,  and 
the  emergent  ray  HI,  till  they  cut  each  other  in  Q.  Let  us  suppose, 
moreover,  that  the  constancy  of  the  ratios  of  refraction  is  true  of  the 
angles  themselves  considered  as  very  small.  Then  the  angle  by 
which  the  ray  is  turned  from  its  course  by  these  refractions,  will  be 

IQR  =  (n  —  1)  CAB  —  (m  —  I)  ABD. 
Demonstration.     At  JP,  G,  H,  erect  the  incident  perpendiculars 
KL,  SM,  JVO,  and  produce  them  till  the  first  cuts  the  second  in  L, 
and  the  second  cuts  the  third  in  JV.     For  the  sake  of  abreviation  let 

CAB  =  A;  ABD  =  B;  EFK  =  F. 

We  first  remark  that  the  angles  formed  by  the  two  incident  per- 
pendiculars, which  fall  upon  each  prism,  are  equal  to  the  refracting 
angle  of  each  of  the  prisms  ;  consequently, 

SLF  =  CAB  =  A;  HJYG  =  ABD  =  B. 

This  has  already  been  demonstrated,  article  594.     Accordingly  we 
have 

LFP  :  LFG  :  :  n  :  1  ;  then  LFG  =  -F: 

n 

L  GF  =  SLF  —  LFG  =  A  —  -F-f 

moreover,  according  to  article  626. 

LGF  :  HGM  :  :  m  :  n  ; 
whence  we  obtain 

-LGF=-A  —  -F; 

m  mm 


NHG  =  HGM  —  HNG  =  -A  —  ±  F—B: 

m  m 

and,  as 

JYHG  :  IHO  ::l  :m, 


Mathematical  Additions.  301 

we  hare 

IHO  =  mNHG  =  nA  —  F—m  B. 

But 

IQR  —  QHP  +  HPQ  =  QHP  +  FGP  +  PFG. 

We  have  also  from  what  precedes, 
(1.)  QHP=IHO—NHG=+nA—F—tnB—-A+-F+B; 


(20 

»«  //*  /« 

(3.)  PFG  =  FFJf  —  £FG  =  +  F  —  -F. 

n 

Consequently, 

IQR  =  (n  —  1)  A  —  (m  —  1)  B. 

628.  Addition.  The  ratios  of  refraction  n  :  1,  w  :  1,  may  be- 
long to  the  mean  rays.  For  the  most  refrangible  violet  rays,  these 
ratios  may  be  JV :  1  and  M  :  1  ;  and  then  the  angle  by  which  the 
violet  ray  is  turned  from  its  primitive  direction,  after  all  the  refrac- 
tons,  will  be 

Now  if  the  emergent  light  is  uncoloured,  the  rays  of  different  col- 
ours are  parallel  to  one  another  after  refraction.  Consequently,  the 
angle  IQR  is  the  same  for  all.  Accordingly,  if  we  make  its  value 
equal  to  that  which  the  mean  rays  give,  we  shall  have  the  equation 


whence 

(jV—  n)  A  —-  (M  —  m}  B, 
or 

M  —  m:  JV—  n::A:B. 

According  to  the  experiments  of  Dollond,  article  619,  crown  glass 
and  flint  glass  produce  an  uncoloured  refraction,  when  A  =  30,  and 
B  =  19.  We  have,  then,  with  respect  to  these  two  substances, 

JV—  n  :M  —  m::19:  30, 
or,  very  nearly, 

JV—  n:M—  n::2:3. 

We  call  the  values  A*  —  n  and  M  —  m,  the  measure  of  the  dis- 
persion of  colours.  This  ratio  has  not  been  hitherto  determined  by 


302  Optics. 

any  general  law.     It  can  only  be  ascertained  in  each  particular  case 
by  direct  experiments,  similar  to  those  of  Dollond. 

629.  Remark.     If  the  experiments  of  Newton,  article  611,  were 
exact,  IQR  must  be  equal  to  0,  as  well  for  the  mean  rays  as  for  the 
violet,  when  the  emergent  light  is  uncoloured.     Then  we  should 
have 

(n— l).*  =  (m-l)B, 
or 

m  —  i  :  n  —  l::A:B. 
Moreover, 

(JV—  l)A  =  (M—  \}B; 
consequently, 

M—l  :JV—  1  ::A:B. 
Whence  it  would  follow 

M  —  1  :  m  —  1  :  :  JV  —  1  :  n  —  I. 

Now  the  difference  between  the  1st  and  2d  term  is  to  the  3d,  as  the 
difference  between  the  3d  and  4th,  is  to  the  4th  ;  that  is, 

M  —  m  :m  —  1  ::JV  —  n  :n  —  1, 
or 

M  —  m:JV  —  n  :  :  m  —  1  :  n  —  1. 

Such  was  the  law  of  Newton ;  it  was  erroneous ;  1 .  Because  it 
was  founded  upon  inexact  observations ;  2.  Because  it  supposed 
that  the  ratio  of  refraction  belongs  to  the  angle  itself,  which  only  ap- 
proaches the  truth  when  the  angles  are  very  small.  When  we  make 
the  calculation  exactly,  without  neglecting  any  thing,  the  experiment 
of  Newton  gives  no  determinate  ratio  (618.) 

630.  The  first  law  of  Euler  was  very  different  from  this ;   he 
thought  that  M  must  depend  upon  m,  in  the  same  manner  as  JV  upon 
n  ;  and  he  showed  that  this  could  only  be  possible  in  the  case  where 
we  have, 

log  M  :  log  m  :  :  log  JV  :  log  n. 

631.  Problem.     In  Jl  (fig.  99)  let  there  be  a  spherical  glass,  for 
which 'the  mean  ratio  of  refraction  is  n  :  1.     Let  a  second  glass  of 
another  substance,  for  which  the  mean  ratio  of  refraction  is  m  :  1 ,  be 
placed  so  near  against  the  other,  that  their  distance  JIB  may  be  con- 
sidered as  nothing.     These  two  glasses  must  be  disposed  so  as  to 


Mathematical  Additions.  303 

have  the  same  axis  AD.  By  this  arrangement  the  images  of  dis- 
tant objects,  seen  by  refraction,  will  be  formed  at  a  certain  determi- 
nate distance  which  will  depend  upon  the  preceding  data. 

Let  the  focal  distance  of  the  first  glass  be  p,  that  of  the  second  q ; 
whatever  be  their  signs, /and  g  are  the  radii  of  the  two  surfaces  of 
the  first  glass,  h  and  i,  the  radii  of  the  two  surfaces  of  the  second. 
According  to  article  570,  (5),  if,  for  the  sake  of  abridgment,  we  make 


we  shall  have 


=  („-!)  a 

Now  let  C  be  the  focus  of  the  first  glass,  which  we  have  desig- 
naied  by  A;  the  image  of  an  object  at  an  infinite  distance,  formed 
by  this  glass,  will  be  in  C. 

This  image  serves  as  an  object  to  the  second  glass,  and  the  image 
produced  by  this  second  glass  B  is  in  D  ;  thus  BD  is  the  quantity 
upon  which  all  the  effects  depend. 

In  the  general  formula,  -  =  —  |  —  (570),  we  must,  for  the  case 

in  question,  write  q  instead  of  p,  and  —  BC  instead  of  «,  since  the 
object  is  at  the  distance  BC  behind  the  glass;  but  as  we  suppose 
AB  —  0,  we  have  BC  =  —  AC  =  —  p;  and  we  find 


or,  if  we  put  for  -  and  -  their  values  found  above, 
P         V 


BD 

Consequently, 

1 

~  (»—  1)  F  +  (M—  1)  H' 

632.  Addition.  Let  the  ratio  of  refraction  for  the  most  refran- 
gible rays  be  JV:  1  in  the  first  glass,  and  M :  1  in  the  second.  Then 
if  E  is  the  place  where  the  image  is  produced  by  the  most  refrangi- 
ble rays,  according  to  article  630,  we  shall  hare 


304  Optics. 

1 


~  (IV—  1)  F  +  (M  —  1)  // 

Now,  if  the  combination  is  to  be  achromatic,  the  images  of  all  the 
the  colours  must  be  united  in  one.  Consequently,  we  shall  have 
BE  =  BD ;  whence  it  follows  that 

(n—i)F+(m—l)H=(JY—l)F+(M—l)H, 


or 

(JV—  n)  F  +  (JW  —  TO)  H  =  0. 
Now,  as  we  have 

=  (n_l)F,  whcnceF  = 


and 


we  obtain 


-  =2  (m  —  1)  II,  whence  H  =  -. : —  ; 


or,  multiplying  by^<y,  and  transposing  the  second  terms. 

(JV—  n)  M—  m 

~  ' 


Whence 

jV_»         M— 


Those  focal  distances  of  the  two  glasses  must,  therefore,  be  in 
this  ratio,  in  order  to  produce  a  single  uncoloured  image. 

633.  Addition.  According  to  the  experiments  of  Dolloml,  tin,- 
ratio  of  refraction  of  the  mean  rays  in  crown  glass  is  1,55  :  1  ;  con- 
sequently n  —  1  =  0,55.  In  flint  glass  this  ratio  is  1,58  :  1  ;  con- 
sequently m  —  1  =  0,58.  The  dispersion  of  colours  in  the  two 
glasses  being  as  19  :  30  ;  we  shall  have 

JV—  n  :M  —  m:  :  19  :  30  ; 
cdnscquently, 

,       iy          so 

v  '.  b  :  :  -  :  --  ; 
0,55  0,58 

that  is, 

p:q::l  :—  1,497  ...... 


Mathematical  Additions. 

The  last  term  of  this  proportion  being  negative,  it  follows  that  the 
lens,  which  is  of  flint  glass,  must  be  diverging. 

Although  this  result  is  very  exact  in  theory,  it  might  not  be  very 
certain  in  practice,  since  Dollond  only  gives  the  ratio  of  dispersion 
19  :  30,  as  an  approximate  one. 

In  practice  it  would  be  necessary  to  enter  into  a  more  difficult 
calculation  ;  that  is,  to  ascertain  the  most  advantageous  dimensions 
of  the  radii/,  g,  h,  i.  At  the  same  time  it  is  obvious  from  the  for- 
mula 

1  n  —  1        n  —  1      1  m  —  1    ,    m  —  1 

p-~7~         ~7";2~~^~         ~i~ 
that  these  radii  admit  of  infinite  variation  for  the  same  focal  distance. 
By  selecting  the  most  suitable  values,  the  aberration  of  sphericity  also 
may,  according  to  the  theory  of  Euler,  be  made  entirely  to  disap- 
pear, or  at  least  become  very  much  diminished ;   so  that  object- 

isses,  calculated  exactly  in  this  manner,  are  exempt  from  two 
causes  of  confusion. 

It  is  in  the  correction  of  this  aberration  of  sphericity,  that  we  have 
the  means  of  giving  proportionally  a  greater  aperture  to  these  object- 
glasses,  than  to  any  simple  lenses  or  mirrors.  This  is  moreover  the 
cause  of  the  perfect  light  in  achromatic  telescopes.  As  to  the  extent 
of  the  field  of  view  it  depends  upon  the  disposition  of  the  eye-glass. 


Elem. 


39 


APPENDIX  TO  OPTICS. 


MANY  optical  phenomena  relating  to  the  physical  properties  of 
light,  having  of  late  years  acquired  some  importance,  we  will  here 
give,  not  a  detailed  account  of  them,  which  would  not  suit  the  plan 
of  this  work,  but  a  sketch  which  will  indicate  the  principal  results. 


Coloured  Rings. 

When  two  plates  of  glass  whose  surfaces  are  not  quite  plane,  are 
placed  one  on  the  other,  the  lamina  of  air  naturally  adhering  to  these 
surfaces  has  usually  thickness  enough  to  exercise  a  complete  action 
on  light,  that  is,  it  reflects  and  refracts  all  the  coloured  rays  in  the 
same  manner  as  if  it  were  of  considerable  depth.  If,  however,  one 
of  the  glasses  be  rubbed  on  the  other,  and  forcibly  pressed  to  it,  to 
exclude  a  part  of  the  intermediate  air,  there  will  soon  be  perceived 
a  degree  of  adhesion,  which  is  generally  greater  in  some  parts  than 
in  others,  either  because  the  surfaces  are  always  a  little  curved,  or 
because  they  invariably  bend  under  strong  pressure ;  in  this  manner 
there  is  obtained  a  lamina  of  air,  thinner  than  the  preceding,  and  the 
depth  of  which  increases  gradually  in  all  directions  from  the  point  in 
which  the  surfaces  are  most  closely  in  contact.  If  now  these  glasses 
be  turned  so  that  the  eye  may  receive  the  light  of  the  clouds,  reflect- 
ed by  the  lamina  of  air,  there  will  be  perceived  a  number  of  concen- 
tric coloured  rings,  which,  when  the  glasses  are  pressed  sufficiently, 
surround  a  dark  spot,  at  the  point  of  contact. 

These  coloured  rings  may  be  formed  by  pressing  together  trans- 
parent plates  of  any  other  substance,  besides  glass ;  they  may 
be  observed,  when  a  glass  lens  is  placed  on  a  plane  surface  of  resin, 
of  metal,  of  metallic  glass,  or  any  other  polished  body.  These  rings 
subsist,  moreover,  in  the  most  perfect  vacuum  that  can  be  produced. 
Neither  is  it  necessary  for  their  formation,  that  the  interposed  lamina 
be  of  air,  nor  that  it  be  contained  between  two  solid  substances ;  a 


Coloured  Rings.  307 

iayer  of  water,  of  alcohol,  of  ether,  or  any  other  evaporable  liquid, 
spread  on  a  black  glass,  produces  similar  colours,  when  sufficiently 
attenuated  ;  they  may  be  observed  also  on  soap  bubbles,  and  on 
blown  glass,  when  thin  enough.* 

In  whatever  manner,  and  under  whatever  circumstances  these 
rings  are  formed,  the  succession  of  their  colours  from  the  central 
dark  spot  is  invariably  the  same  ;  the  only  difference  perceptible  is 
in  their  brightness,  which  varies  with  the  refracting  power  of  the 
lamina,  and  in  their  form,  which  depends  on  the  law  by  which  the 
thickness  of  the  lamina  is  regulated  in  different  parts.  In  fact,  for 
any  one  substance,  the  colour  reflected  at  any  point  depends  on  the 
thickness  of  the  lamina,  and  the  incidence  under  which  the  reflection 
takes  place. 

So  far  we  have  supposed  the  colours  of  the  lamina  to  be  seen 
only  by  reflection  ;  if  it  be  placed  between  the  eye  and  the  light, 
concentric  rings  will  again  be  observed  similar  to  the  others  in  form, 
but  not  in  colour,  and  fainter,  surrounding  a  bright  spot. 

This  might  naturally  be  expected,  for  when  the  incident  light  is 
decomposed,  so  as  to  give  coloured  rays  in  the  reflection,  those 
transmitted  must  of  course  be  also  coloured,  and  the  one  set  must, 
in  fact,  be  complementary  to  the  other,  that  is,  both  together  would 
produce  white. 

It  follows  from  all  this,  that  to  discover  the  laws  of  these  phe- 
nomena, the  best  method  is  to  study  them  in  cases  where  the 
variation  of  thickness  is  regular  and  known.  This  is  what  Newton 
did ;  and  he  conducted  his  researches  with  a  careful  nicety,  which 
could  be  owing  only  to  the  importance  which  he  foresaw  would  be 
attached  to  the  consequences  of  them. 

He  formed  the  rings  by  placing  a  convex  glass  of  small  curvature 
on  a  piece  of  perfectly  plane  glass  ;  then  the  thickness  of  the  lamina 
of  air  increasing  symmetrically  in  all  directions  from  the  point  of 
contact,  the  rings  were  perfectly  circular  round  the  dark  spot  formed 
at  that  point. 

He  measured  the  diameters  of  these  rings,  in  a  particular  case, 
and  thence,  knowing  the  curvature  of  the  surface,  he  was  able  to 
calculate  the  thickness  of  the  lamina  at  each  ring. 

Repeating  this  observation  under  different  angles  of  incidence,  he 

*  Of  the  same  nature  are  the  coloured  stripes  often  seen  in  cracked 
ice,  in  transparent  calcareous  spar,  selenite,  and  other  substances. 


308  Optics. 

remarked  the  variations  produced  in  the  rings ;  he  found  that  they 
grew  wider  as  the  obliquity  increased,  and  by  measuring  their  diam- 
eters, he  calculated  the  different  thicknesses  at  which  the  same  col- 
our appeared. 

He  made  similar  experiments  on  thin  plates  of  water,  contained 
between  two  glasses,  and  on  thin  soap  bubbles,  blown  with  a  pipe. 
These  bubbles  being  placed  on  a  plane  glass,  became  perfectly 
hemispherical,  and  being  covered  over  with  a  bell-glass,  they  lasted 
long  enough  for  him  to  observe  at  leisure  their  brilliant  tints.  He 
thus  found  that  the  thicknesses,  at  which  the  same  colours  appeared 
were  less  than  in  air,  in  the  ratio  of  3  to  4,  which  is,  in  fart,  that  of 
refraction  between  these  two  substances.  Other  trials  with  laminae 
of  glass,  led  him  to  generalize  this  remark,  which  many  other  experi- 
ments afterwards  confirmed.  He  collected  all  his  results  into  empiric 
tables,  which  express  the  laws  of  them  in  numbers. 

These  laws  were,  however,  still  complicated  in  consequence  of 
the  unequal  refrangibilities  of  the  different  rays,  by  which  the  rings 
were  illuminated.  To  reduce  the  phenomenon  to  its  greatest  sim- 
plicity, Newton  formed  rings  with  simple  light,  by  looking,  in  a  dark 
room  at  a  white  paper,  which  received  in  turns  all  the  simple  col- 
ours of  the  prismatic  spectrum.  This  paper  thus  enlightened,  and 
seen  by  reflection  on  the  thin  lamina?,  became  like  a  kind  of  sky, 
coloured  by  that  tint  alone,  which  was  thrown  on  it.  In  this  manner 
the  following  results  were  obtained  ; 

(1.)  Each  kind  of  simple  light  produced  rings  of  its  own  colour, 
both  by  reflection  and  by  transmission. 

(2.)  In  each  case,  the  rings  were  separated  by  dark  intervals, 
which  made  them  much  more  distinct  than  in  the  original  experi- 
ment, and  caused  many  more  to  be  discerned.  They  were  more 
and  more  crowded  together  as  their  distance  increased  from  the 
central  spot. 

(3.)  The  dark  intervals  which  separated  the  bright  rings,  seen  by 
the  reflected  light,  were  bright  rings  themselves  by  the  transmitted 
rays,  and  they  were  separated  by  dark  intervals  answering  to  the 
former  rings.  However,  those  intervals  were  not  exactly  black,  be- 
cause the  reflection  on  a  thin  lamina  of  air  is  far  from  being  perfect, 
even  in  the  most  brilliant  part  of  the  reflected  rings ;  and  the  same 
tiling  may  be  observed  of  all  thin  transparent  plates  of  any  substance 
whatever. 


Coloured  Rings. 

(4.)  In  observing  the  luminous  reflected  rings,  Newton  remarked, 
that  they  were  not  simple  geometrical  lines,  but  that  each  of  them 
occupied  a  certain  space,  in  which  the  brightness  diminished  gradu- 
ally each  way  from  the  middle. 

(5.)  Measuring  the  diameters  of  the  reflected  rings  at  their  bright- 
est part,  he  found  that  for  each  particular  kind  of  rays,  the  squares 
of  the  diameters  followed  the  arithmetical  progression  of  the  num- 
bers 1 ,  3,  5,  7,  &c. ;  consequently,  the  thicknesses  of  the  lamina, 
which  are  as  the  squares  of  their  diameters,  were  in  that  same  pro- 
gression. 

When  the  glasses  were  illuminated  by  the  brightest  part  of  the 
spectrum,  which  is  between  the  orange  and  yellow,  the  diameter  of 
the  sixth  ring  was  found  to  be  the  same  as  that  of  the  brightest  part 
of  the  corresponding  ring  in  the  experiment  made  in  full  day-light. 

(6.)  The  diameters  of  the  dark  rings  being  likewise  measured,  he 
found  that  their  squares,  and  consequently,  the  thicknesses  of  the  air 
below  them,  followed  in  the  progression,  2,  4,  6,  8,  &c. 

(7.)  By  other  measurements,  he  discovered  that  the  brightest 
parts  of  the  transmitted  rings  answered  to  the  darkest  parts  of  the 
intervals  in  reflection,  and  vice  versa,  the  darkest  parts  here  were  the 
brightest  in  the  other  case,  so  that  the  thicknesses  of  air  which  t»x1ns- 
mitted  the  bright  rings,  and  those  which  gave  dark  inJgfvaTs,  were 
respectively  as  2,  4,  6,  8,  &c.,  and  as  1,  3,  5,  7,  §^. 

(8.)  The  absolute  diameters  of  corresponding  rings  of  different 
colours  were  different,  as  were  also  their  breadths,  both  these  dimen- 
sions being  greatest  for  the  extreme  red  rays,  and  least  for  the  violet. 

(9.)  The  simple  rings  of  each  colour  were  least  when  the  rays 
passed  perpendicularly  through  the  lamina  of  air,  and  increased  with 
the  angle  of  incidence. 

These  observations  explain  completely  the  more  complicated  phe- 
nomenon of  the  rings  formed  by  the  natural  light ;  for  this  light,  con- 
sisting of  different  coloured  rays  mixed  together  in  definite  propor- 
tions, when  a  beam  of  this  mixture  falls  on  the  thin  lamina  of  air 
between  the  glasses,  each  kind  of  simple  light  forms  its  own  rings  by 
itself,  according  to  its  own  peculiar  laws,  and  as  the  diameters  of 
these  rings  are  different  for  the  various  kinds  of  light,  they  are  suffi- 
ciently separated  from  each  other  to  be  distinguished.  However, 
this  separation  is  by  no  means  so  perfect  as  in  observations  made 
with  simple  rings,  because  the  rings  of  different  colours  encroach  a 
little  on  each  other,  so  as  to  produce  that  infinite  diversity  of  tints 


3i  Optics. 

that  the  t  .it  shows.     But,  though  this  successive  superposi-  - 

tion  of  the  s.       e  rings  is  really  the  key  of  the  phenomena,  one  can- 
not be  very  sure  of  the  fact  without  having  measured  exactly  the 
absolute  magnitudes  of  the  diameters  and  breadths  of  the  rings, 
formed  by  the  different  coloured  rays ;  for  when  these  results  are 
once  known,  it  can  only  be  a  simple  arithmetical  problem  to  find  the 
species  and  the  quantity  of  each  colour  that  may  be  reflected  or 
transmitted  at  each  determinate  thickness  ;  and  consequently,  if  the 
effects  of  the  composition  of  all  these  colours  be  calculated  by  the 
rules  which  Newton  has  given  in  his  Optics,  it  will  be  easy  to  deduce 
with  perfect  accuracy,  the  numerical  expressions  of  the  tint  and  in- 
tensity of  colour  that  must  exist  at  each  point  of  the  compound 
rings,  which  may  then  be  compared  with  experiment.     In  a  wor 
we  have  as  yet  only  a  suspicion,  a  probable  one  no  doubt,  of 
cause  of  our  phenomena  ;  accurate  measurements  are  necess- 
convert  that  probability  into  certainty. 

This  is  just  what  Newton  did.     He  measured  the  dia- 
*^e  simple  rings  of  the  same  order,  both  at  their  inner 
'-:nor  successively  the  various  colours  of  the  spr 
^  the  deepest  red ;  afterwards,  ar 
-onnect  these  results  f 

'*'»  sufficient  en 

compai ..  ^  onal 

thickness  of  tlib  .  rings. 

Similar  measurements,  eiitx.  /ders  of 

rings,  formed  by  one  simple  coknu,  .he  inter- 

vals of  thickness,  throughout  which  reflect,.  were  sen- 

sibly equal  to  those  which  allowed  transmission,  i  i  when  the 
light  was  incident  perpendicularly.  Thus,  designating  generally  by 
t  the  thickness  of  the  air  at  the  beginning  of  the  first  lucid  ring,  for 
any  simple  colour,  that  ring  ended  at  the  thickness  3  t,  and  therefore 
occupied  an  interval  of  thickness  equal  to  2 1.  Then  came  the 
first  dark  ring,  occupying  an  equal  interval  2  t ;  then  a  second  lucid 
ring  from  5  t  to  7  t,  and  so  on. 

Combining  this  law  of  succession  for  the  different  orders,  with  that 
of  the  distribution  of  the  various  tints  of  the  same  order,  one  easily 
conceives  that  a  single  absolute  thickness,  measured  at  the  beginning, 
the  middle,  or  the  end  of  any  ring,  formed  by  a  simple  colour,  is 
sufficient  to  calculate  the  value  of  the  first  thickness  *,  relatively  to 
that  colour,  and  thus  all  the  thicknesses  of  the  several  rings  of  each 
colour  may  be  determined. 


Coloured  Rings. 


311 


In  this  manner  Newton,  measuring  the  thickness  represented  by 
2  t  for  the  different  simple  rays,  in  vacuo,  in  air,  in  water,  and  in 
common  ghss,  found  their  values  as  shown  in  the  following  table, 
where  they  are  expressed  in  ten  thousandth  parts  of  an  inch. 


Colours. 

Values  of  2  t. 

In  Vacuo. 

In  Air. 

In  Water. 

In  Glass 

E 

'o 

E 

3,99816 
4,32436 
4,51475 

4,84284 
5,23886 
5,61963 
5,86586 
6.34628 

3,99698 
4,32308 
4,51342 
4,84142 
5,23732 
5,61798 
5,86414 
6.34441 

2,99773 
3,24231 
3,38507 
3,63107 
3,92799 
4,21349 
4,39811 
4.75S3T 

2,57870 
2,78908 
2,91188 
3,12350 
3,37891 
3,62450 
3,78331 
4.003  1  7 

r  violet  and  indigo.... 
indigo  and  blue  
blue  and  green  
green  and  yellow... 
yellow  and  orange., 
^orange  and  red  

In  this  table  the  values  relating  to  air  were  alone  immediately  ob- 
tained by  observation ;  the  others  were  calculated  from  the'm  by 
means  of  the  several  ratios  of  refraction,  that  is,  by  multiplying  them 
by  |f  f  |  for  the  vacuum,  £  for  water,  and  f  f  for  glass.  It  must 
be  remembered,  that  these  values  all  suppose  the  incidence  to  be 
perpendicular. 

Applying  to  these  results  a  rule  that  he  had  found  to  determine 
the  nature  of  the  compound  colour  resulting  from  any  given  mixture 
of  simple  colours,  Newton  deduced  the  following  table,  which  shows 
the  thickness  at  which  the  brightest  tints  of  each  ring  appear,  when 
seen  under  the  perpendicular  incidence.  This  table  is  calculated 
only  for  air,  water,  and  common  glass,  but  may  of  course  be  extend- 
ed to  all  other  substances,  by  the  method  above  mentioned. 

The  unit  is  the  thousandth  part  of  an  inch.  By  the  side  of  differ- 
ent colours  are  put  the  names  of  certain  flowers  or  metallic  sub-- 
stances, just  to  give  more  distinct  ideas  of  them. 


312 


Optics. 


Colours  reflected. 

Thicknesses  in  thousandth 
parts  of  an  inch. 

Names  of  the  Colours,  or 
substances  having  them. 

in  air. 

in  water. 

In  glass. 

1st  Order. 
Very  black 
Black 

I2 

2 

2f 

n 

8 
9 

H 

3J 

6* 

c  -  ;  -  -|0 

-|n  c»|n  f\tf  -l«  o»l"5  "K->  -l»  *l« 
—  ~  CO  Tf  1C  0 

Whitish  sky-blue. 
Tarnished  silver. 
Straw  colour. 
Dried  orange-peel. 
Geranium  Sanguineum. 

Beginning  of  black 
Blue 

White  

Yellow 

Orange  . 

Red    

2d  Order. 
Violet  .  .      .  . 

u! 

14* 
15J 

in 

18* 

10t 

135 

133 

143 

91T 

9! 

Iodine. 
Indigo. 
Cobalt  blue. 
Water,  aquamarine. 
Lemon. 
3range. 
Bright  May-pink. 

Indigo 

Blue  

Green             . 

Yellow  

Orange  

Bright  red   .... 
Scarlet  

3rd  Order. 
Purple  .... 

21 

23f* 

2?t 
291 
32 

153 

l?f, 

201* 

213 

24 

wj* 
!il 

20| 

Flax-blossom. 
Indigo. 
Prussian  blue. 
Bright  meadow  green. 
White  wood. 
Rose. 

Blue  

Green 

Yellow  

Red 

Bluish  red    .... 
4th  Order. 
Bluish  green  .  .  . 
Green          . 

34 
85f 

36 
401 

26| 
27 
301 

22 
223 

26 

Emerald. 
Pale  pink. 

Yellowish  green  .  . 
Red    . 

5th  Order. 
Greenish  blue    .  . 
Red    

46 
521 

341 
39  f 

29  f 
34 

Sea-green. 
Pale  pink. 

6th  Order. 
Greenish  blue  .  .  . 
Red    

583 
65 

44 

48| 

38 
42 

Ijijrht  sea-green. 
Paler  red. 

7th  Order. 
Greenish  blue  .  .  . 
Ruddy  white  .  .  . 

71 

77 

531 
57f 

45i 
49| 

f  cry  faint. 
Ditto. 

Coloured  Rings.  313 

Reduction  of  the  phenomena  of  the  rings  to  a  physical  property  of 
light,  called  fits  of  easy  reflection  and  transmission. 

The  phenomena  of  the  rings  being  reduced  to  laws  extremely 
exact  and  well  adapted  to  calculation,  Newton  concentrated  them  all 
in  a  still  simpler  expression,  making  them  depend  on  a  physical  pro- 
perty, which  he  attributed  to  light,  and  of  which  he  defined  all  the 
particulars  conformably  to  their  laws. 

Considering  light  as  a  matter  composed  of  small  molecules  emitted 
by  luminous  bodies  with  very  great  velocities,  he  concluded,  that 
since  they  were  reflected  within  the  lamina  of  air,  at  the  several 
thicknesses  t,  3 t,  5  t,  7 1,  &c.,  and  transmitted  at  the  intermediate 
thicknesses  0,  2  t,  4 1,  6 1,  &tc.,  the  molecules  must  have  some  peculiar 
modification  of  a  periodical  nature,  such  as  to  incline  them  alter- 
nately to  be  reflected  and  refracted  after  passing  through  certain 
spaces.  Yet  this  modification  could  not  be  necessary,  since  the  in- 
tensity of  the  reflection  at  the  second  surface  varies  with  the  medium 
contiguous  to  that  surface,  so  that  a  given  molecule  arriving  at  it,  at 
a  given  epoch  of  its  period,  may  be  either  reflected  or  transmitted, 
according  to  the  exterior  circumstances  which  act  on  it.  Newton, 
therefore,  characterized  this  property  of  the  luminous  molecules  as  a 
simple  tendency,  and  designated  it  appropriately  enough  by  the 
phrase,  jif  of  easy  rejection,  or  transmission. 

According  to  this  idea  of  the  fas,  their  duration  must  evidently  be 
proportional  to  the  thickness  t,  which  regulates,  in  each  substance, 
the  alternations  of  reflection  and  transmission.  Thus,  in  the  first 
table  given,  we  find  the  measure  of  it  for  a  vacuum,  for  air,  water, 
and  glass,  in  the  case  of  perpendicular  incidence.  In  other  sub- 
stances, the  duration  of  the  fits  must  vary  as  the  quantity  t,  that  is, 
inversely  as  the  refracting  power ;  it  will  vary  also,  by  parity  of  rea- 
son, with  the  obliquity  of  incidence,  and  the  nature  of  the  light ;  but 
the  laws  of  these  variations  are  exactly  those  which  regulate  the 
rings  themselves ;  so  that,  these  last  being  known,  it  remains  only  to 
apply  them ;  this  Newton  did,  and  after  having  defined  completely 
all  the  characters  of  the  fits,  he  employed  them  as  a  simple  property, 
not  only  to  unite  under  one  point  of  view  the  phenomena  of  the  col- 
ours produced  by  thin  plates,  but  also  to  foresee  and  to  calculate 
beforehand,  both  as  to  their  general  tenor,  and  their  minutest  details, 
a  crowd  of  analogous  phenomena  observed  to  attend  reflection  in 
thick  plates,  which,  in  fact,  in  his  experiments,  exceeded  by  as  much 
as  twenty  or  thirty  thousand  times  those  on  which  the  calculations 
Elem.  40 


314  Optics. 

had  been  founded  j  moreover,  applying  the  same  reasoning  to  the 
integrant  particles  of  material  substances,  which  all  chemical  and 
physical  phenomena  show  to  be  very  minute,  and  to  be  separated, 
even  in  the  most  solid  bodies,  by  spaces  immense  in  comparison  of 
their  absolute  dimensions,  he  was  able  to  deduce  naturally  from  the 
same  principles,  the  theory  of  the  different  colours  they  present  to 
us,  a  theory  which  adapts  itself  with  a  surprising  facility  to  all  the 
observations  to  which  those  colours  can  be  submitted.  The  number 
and  importance  of  those  applications  account  sufficiently  for  the  care 
which  Newton  took  with  his  experiments  on  the  rings ;  I  am  sorry 
to  be  obliged  to  confine  myself  here  to  the  bare  indication  of  those 
fine  discoveries. 


Another  Explanation  of  the   Coloured  Rings  on  the  Hypothesis  of 
Undulations. — Dr  Young's  principle  of  Interferences. 

If  light  be  really  a  material  substance,  Newton's /te  are  a  neces- 
sary property,  because  they  are  only  a  literal  enunciation  of  the 
alternations  of  reflection  and  transmission  which  coloured  rings  pre- 
sent ;  but  if  light  be  otherwise  constituted,  these  alternations  may  be 
accounted  for  differently. 

Descartes,  and  after  him  Huygens,  and  a  great  number  of  natural 
philosophers,  have  supposed  that  the  sensation  of  light  was  produced 
in  us  by  undulations  excited  in  a  very  elastic  medium,  and  propa- 
gated to  our  eye,  which  they  affect  in  the  same  manner  as  undula- 
tions excited  in  the  proper  medium  of  the  air,  and  propagated  to  the 
ear,  produce  in  it  the  sensation  of  sound.  This  medium,  if  it  does 
exist,  must  fill  all  the  expanse  of  the  heavens,  since  it  is  through  this 
expanse  that  the  light  of  the  stars  comes  to  our  eyes  ;  it  must  also 
be  extremely  elastic,  since  the  transmission  of  light  takes  place  with 
such  extraordinary  velocity  ;  and  at  the  same  time  its  density  must 
be  almost  infinitely  small,  since  the  most  exact  discussion  of  ancient 
and  modern  astronomical  observations,  does  not  indicate  the  least 
trace  of  resistance  in  the  planetary  motions.  As  to  the  relations  of 
this  medium  with  earthly  bodies,  it  is  plain  that  it  must  pervade  them 
all,  for  they  all  transmit  light  when  sufficiently  attenuated  ;  moreover, 
its  density  must  probably  differ  in  them  according  to  the  nature  of 
the  substances,  since  unequal  refractions  appear  to  prove  that  the 
propagation  of  light  takes  place  in  different  media  with  various  velo- 


Coloured  Rings.  315 

cities.  But  what  ought  to  be  the  proportions  of  the  densities  for 
these  different  substances  ?  How  is  the  luminiferous  ether  brought  to, 
or  kept  in,  the  proper  state  for  each  ?  How  is  it  inclosed  and  con- 
tained so  as  to  be  incapable  of  spreading  out  of  them  ?  Moreover, 
how  is  this  medium,  so  non-resisting,  so  rare,  so  intangible,  agitated 
by  the  molecules  of  bodies  which  appear  to  us  luminous  ?  These 
are  so  many  characters  which  it  would  be  necessary  to  know  well, 
or  at  least  to  define  well,  to  have  an  exact  idea  of  the  conditions  ac- 
cording to  which  the  undulations  are  formed  and  propagated  ;  but 
hitherto  they  have  never  been  distinctly  established. 

At  any  rate,  if  a  body  be  conceived  to  have  the  faculty  of  exciting 
an  instantaneous  agitation  in  a  point  of  such  a  medium,  supposed  at 
first  equally  dense  in  all  its  extent,  this  agitation  will  be  propagated 
in  concentric  spherical  waves,  in  the  same  manner  as  in  air,  except 
that  the  velocity  will  be  much  more  considerable.  Each  molecule 
of  the  medium  will  then  be  agitated  in  its  turn,  and  afterwards  return 
to  a  state  of  rest. 

If  these  agitations  are  repeated  at  the  same  point,  there  will  result, 
as  in  air,  a  series  of  undulations  analogous  to  those  producing  sound  ; 
and  as  in  these  there  are  observed  successive  and  periodical  alterna- 
tions of  condensation  and  rarefaction,  corresponding  to  the  alterna- 
tions of  direction  which  constitute  the  vibrations  of  a  sonorous  body, 
in  like  manner,  it  will  be  easily  conceived  that  the  successive  and 
periodical  vibrations  of  luminous  bodies  might  produce  similar  effects 
in  luminous  undulations  ;  and  again,  as  the  succession  of  sonorous 
waves,  when  sufficiently  rapid,  produces  on  our  ear  the  sensation  of 
a  continuous  sound,  the  quality  of  which  depends  on  the  rapidity  of 
the  opposite  vibrations,  and  on  the  laws  of  condensation  and  velocity 
that  the  nature  of  these  vibrations  excites  in  each  sonorous  wave  ;  in 
like  manner,  under  analogous  conditions,  the  ethereal  waves  may 
produce  sensations  of  light  in  our  eyes,  and  different  sensations  in 
consequence  of  the  variety  of  the  conditions.  Hence  the  differences 
of  colour.  In  this  system,  the  length  of  the  luminous  waves  cor- 
respond to  Newton's  ft^  and  their  length  is,  as  will  be  seen  here- 
after, exactly  quadruple  ;  the  rapidity  of  their  propagation  depends, 
as  in  air,  on  the  relation  between  the  elastic  force  of  the  fluid  and 
its  density. 

When  a  sonorous  wave  excited  in  air  arrives  at  the  surface  of  a 
solid  body,  its  impact  produces  in  the  parts  of  that  body  a  motion, 
insensible  indeed,  but  nevertheless  real,  which  sends  it  back.  If  the 


316  Optics. 

body,  instead  of  being  solid,  is  of  a  gaseous  form,  the  reflection  takes 
place  equally,  but  there  is  produced  in  the  gas  a  sensible  undulation 
depending  on  the  impression  that  its  surface  has  received.*  Lumi- 
nous undulations  ought  to  produce  a  similar  effect  when  the  medium 
in  which  they  are  excited  is  terminated  by  a  body  in  which  the  den- 
sity of  the  ethereal  fluid  is  different ;  that  is  to  say,  there  must  be 
produced  a  reflected  wave  and  one  transmitted  ;  which  is,  in  fact, 
what  we  call  reflection  and  refraction.  In  this  system,  the  intensi- 
ties of  rays  of  light  must  be  measured  by  the  vw  viva  of  the  fluid  in 
motion,  that  is,  by  the  product  of  the  density  of  the  fluid  by  the 
square  of  the  proper  velocity  of  its  particles. 

To  confirm  these  analogies,  already  very  remarkable,  it  would  be 
necessary  to  follow  up  their  consequences  further  with  calculations  ; 
but  unfortunately  this  cannot  be  done  rigorously.  The  subject  of 
undulations  thus  sent  back  or  transmitted  in  oblique  motions,  is  be- 
yond the  existing  powers  of  analysis.  In  the  case  of  perpendicular 
incidence  the  phenomenon  becomes  accessible,  but  then  it  teaches 
nothing  as  to  the  general  direction  of  the  motion  communicated,  as 
the  propagation  must  be  continued  in  a  straight  line,  if  for  no  other 
reason,  on  account  of  there  being  no  cause  why  it  should  deviate 
from  it ;  nevertheless  in  this  case  theory  indicates  the  proportions  of 
intensity  for  the  incident  and  reflected  waves,  which  appear,  in  fact, 
tolerably  conformable  to  experiments  on  light,  which  is,  at  any  rate, 
a  verification  as  far  as  it  goes. 

When  the  ear  receives  at  once  two  regular  and  sustained  sounds, 
it  distinguishes,  besides  those  sounds,  certain  epochs  at  which  undu- 
lations of  the  same  nature  arrive  together  or  separate.  If  the  periods 
of  these  returns  are  very  rapid,  a  third  sound  is  heard,  the  tone  of 
which  may  be  calculated  a  priori  from  the  epochs  of  coincidence  ; 
but  if  these  happen  so  seldom  as  to  be  heard  distinctly,  and  counted, 
the  effect  is  a  series  of  beats  which  succeed  each  other  more  or  less 
rapidly.  The  mixture  of  two  rays,  which  arrive  together  at  the  eye, 
under  proper  circumstances,  produces  an  effect  of  the  same  kind, 


*  This  phenomenon  may  be  observed  in  the  sounds  produced  by 
organ-pipes  when  filled  with  successive  strata  of  gases  of  unequal 
densities,  for  instance,  with  atmospheric  air  and  hydrogen.  The 
sounds  which  should  be  produced  under  such  circumstances,  have 
been  calculated  by  Mr  Poisson,  and  his  results  agree  perfectly  with 
experiment. 


Coloured  Rings.  317 

which  Grimaldi  remarked  long  ago,  but  of  which  Dr  Young  first 
showed  the  numerous  applications.  The  neatest  way  of  exhibiting 
this  phenomenon  is  the  following,  which  is  due  to  M.  Fresnel. 

A  beam  of  sun-light,  reflected  into  a  fixed  direction  by  a  heliostat, 
is  introduced  into  a  darkened  room  ;  it  is  transmitted  through  a  very 
powerful  lens,  which  collects  it  almost  into  a  single  point  at  its  focus. 
The  rays  dive  ^ing  from  thence  form  a  cone  of  light  within  which 
there  are  placed,  at  the  distance  of  two  or  three  yards,  two  metallic 
mirrors  inclined  to  each  other  at  a  very  small  angle,  so  that  they 
receive  the  rays  almost  under  the  same  angle  ;  the  observer  places 
himself  at  a  certain  distance,  so  as  to  observe  the  reflection  of  the 
luminous  point  in  both  the  mirrors.  There  are  thus  seen  two  images 
separated  by  an  angular  interval  which  depends  on  the  inclination  of 
the  two  mirrors,  their  distances  from  the  radiating  point,  and  the 
place  of  the  observer  ;  but  besides  these,  which  is  the  essential  point 
of  the  phenomenon,  there  may  be  seen,  by  the  help  of  a  strong 
magnifying  lens,  between  the  places  of  the  two  images,  a  series  of 
luminous  coloured  fringes  parallel  to  each  other,  and  perpendicular 
to  the  line  joining  the  images  ;  if  the  incident  light  is  simple,  the 
fringes  are  of  the  colour  of  that  light,  and  separated  by  dark  inter- 
vals. Their  direction  depends  solely  on  those  of  the  mirrors,  and 
not  at  all  on  any  influence  of  the  edges  of  those  mirrors,  as  each  of 
them  may  be  turned  round  in  its  own  plane  without  producing  the 
slightest  alteration  in  the  phenomenon. 

Let  us  confine  our  attention,  for  the  sake  of  greater  simplicity,  to 
the  case  in  which  the  incident  light  is  homogeneous  ;  this  case  may 
be  easily  exhibited  in  practice  by  observing  the  fringes  through  a  col- 
oured glass  which  will  transmit  only  the  rays  of  a  particular  tint.  In 
this  case  if  we  select  any  one  of  the  brilliant  fringes  formed  between 
the  two  images,  we  may  calculate  the  directions  and  paths  of  the 
luminous  rays  which  form  that  fringe,  coming  from  each  of  the  mir- 
rors. Now  in  making  this  calculation,  we  find  the  following  results. 

(1.)  The  middle  of  the  space  comprised  between  the  two  lumi- 
nous points,  is  occupied  by  a  band  of  colour  formed  by  rays  the 
lengths  of  whose  paths  from  the  luminous  point  to  the  eye  are  equal. 

(2.)  The  first  fringe  on  each  side  of  this  is  formed  by  rays  for 
which  the  difference  of  length  is  constant,  and  equal  for  instance  to  /. 

(3.)  The  second  coloured  fringe  arises  from  the  rays  having  2  / 
for  the  difference  of  the  distances  they  pass  over. 


316  Optics. 

(4.)  In  general,  for  each  fringe  this  difference  is  one  of  the  terms 
of  the  series  0,  /,  2  /,  3  I,  4  /,  &c. 

(5.)  The  intermediate  dark  spaces  are  formed  by  rays  for  which 
the  differences  are  ??,$/,£/,  &c. 

(6.)  Lastly,  the  numerical  value  of  7  is  exactly  four  times  that  of 
the  length  which  Newton  assigns  to  the  fa  for  the  particular  kind  of 
light  considered. 

The  analogy  between  these  laws  and  those  of  the  rings  is  evident. 
The  following  is  the  explanation  given  of  them  in  the  system  of  un- 
dulations ;  the  interval  /  is  precisely  equal  to  the  length  of  a  lumi- 
nous wave,  that  is,  to  the  distance  of  those  points  in  the  luminiferous 
ether,  which,  in  the  succession  of  the  waves,  are  at  the  same  mo- 
ment in  similar  situations  as  to  their  motion.  When  the  paths  of  two 
rays  which  interfere  with  one  another,  differ  exactly  by  half  this 
quantity  at  the  place  where  they  cross,  they  bring  together  contrary 
motions  of  which  the  phases  are  exactly  alike.  Moreover,  the  mo- 
tions produced  by  these  partial  undulations  take  place  almost  along 
the  same  line,  as  the  mutual  inclination  of  the  mirrors  is  supposed 
to  be  very  small.  Consequently,  the  two  motions  destroy  one  an- 
other, the  point  of  ether  at  which  they  meet  remains  at  rest,  and  the 
eye  receives  no  sensation  of  light.  The  same  thing  must  occur  at 
those  points  where  the  differences  of  the  spaces  passed  over  by  the 
rays  is  f  ?,  f  I,  or  any  other  such  number  ;  whereas  at  points  where 
the  difference  is  /,  2Z,  3/,  or  any  other  multiple  of  /,  the  undulating 
motions  coincide,  and  assist  each  other,  so  that  the  appearance  of 
light  is  produced. 

This  way  of  considering  the  combination  of  luminous  waves  and 
the  alternations  of  light  and  darkness  which  result  from  it,  has  been 
called  by  Dr  Young,  the  principle  of  interferences. 

The  phenomenon  of  the  alternations  of  light  and  darkness  is  cer- 
tain ;  if,  reasoning  a  priori,  it  appeared  to  be  possible,  only  on  the 
hypothesis  of  undulations,  it  would  reduce  the  probability  of  that  hy- 
thesis  to  a  certainty,  and  completely  set  aside  the  theory  of  emission. 
It  does  not,  however,  appear  to  offer  that  character  of  necessary 
truth  which  would  be  so  valuable,  whichever  argument  it  favoured, 
because  it  would  be  decisive.  One  may,  without  violating  any  rule 
of  logic,  conceive  equally  the  principle  of  interferences  in  the  sys- 
tem of  emissions,  making  the  result  which  it  expresses  a  condition  of 
vision. 


Coloured  Rings.  3jg 

In  fact,  the  phenomenon  of  the  fringes  does  not  prove  that  the 
rays  of  light  really  do  affect  each  other  under  certain  circumstances, 
it  only  shows  that  the  eye  does,  or  does  not  receive  the  sensation  of 
light,  when  placed  at  a  point  where  the  rays  coincide  with  those 
circumstances ;  it  proves  also  that  an  unpolished  surface  placed  at 
such  a  point,  and  seen  from  a  distance,  appears  either  bright  or 
dark ;  now  in  the  former  case  it  is  possible  that  vision  may  cease 
when  the  retina  receives  simultaneously  rays  which  are  at  different 
epochs  of  their  fits  ;  and  in  the  latter,  when  such  rays  arrive  together 
at  an  unpolished  surface,  and  are  afterwards  dispersed  by  radiation 
in  all  directions,  it  is  clear,  that  having  the  same  distance  to  pass 
over  from  each  of  the  surfaces  to  the  eye,  they  will  have,  on  arriving 
at  it,  the  same  relative  phases  that  they  had  when  at  the  surface  ;  if, 
therefore,  they  were  then  in  opposite  states,  they  will  be  so  likewise 
in  arriving  at  the  retina,  and  thus  there  will  be  no  vision  produced. 
I  do  not  pretend  that  this  explanation  is  the  true  one,  or  even  that  it 
bears  the  character  of  necessity  ;  it  is  both  true  and  necessary  if 
light  be  material,  for  it  is  but  the  statement  of  a  phenomenon  ;  but 
if  only  it  implies  no  physical  contradiction,  that  is  quite  sufficient  to 
prevent  the  phenomenon  from  which  it  is  derived  from  being  deci- 
sive against  the  system  of  emission. 

Dr  Young  has  with  equal  success  applied  the  principle  of  inter- 
ferences to  the  explanation  of  the  coloured  rings,  both  reflected  and 
transmitted,  of  thin  plates.  When  such  a  plate  is  seen  by  reflection, 
the  light  coming  from  the  first  surface  to  the  eye  interferes  with  that 
from  the  second  ;  this  interference  either  does  or  does  not  produce 
the  sensation  of  light,  according  as  the  different  distances  that  the 
rays  have  to  pass  over,  place  them  in  similar  or  opposite  phases  of 
their  undulations  ;  but  then,  at  the  point  where  the  thickness  is  noth- 
ing, this  difference  is  nothing,  and  consequently  one  would  expect  to 
see  a  bright  spot  instead  of  a  dark  one.  To  get  over  this  difficulty  Dr 
Young  introduces  a  new  principle,  namely,  that  the  reflection  within 
the  plate  makes  the  rays  lose  an  interval  \  I,  exactly  equal  to  half  the 
length  of  a  wave.  By  means  of  this  modification,  the  rays  reflected 
from  the  two  surfaces  at  the  point  where  the  thickness  is  nothing,  ac- 
quire opposite  dispositions,  and  therefore  produce  together  no  sensation 
of  light  in  the  eye  ;  then  in  the  surrounding  places,  the  law  of  the  pe- 
riods of  the  undulations  gives  that  of  the  succession  of  blight  and  dark 
rings  ;  this  law,  thus  modified,  agrees  with  the  measurements  of  the 
coloured  rings  observed  in  the  case  of  perpendicular  incidence  ;  but 


320  Optics. 

for  oblique  incidences  it  is  not  quite  consistent  with  Newton's  state- 
ment. Is  it  possible  that  the  laws  which  Newton  established  upon 
experiments  may  be  inexact,  or  must  we  introduce  in  the  case  of 
oblique  waves  some  modification  depending  on  their  impact  on  the 
surfaces?  This  point  is  yet  to  be  decided. 

We  have  hitherto  considered  only  the  rings  observed  by  reflected 
light ;  the  others  are  formed,  according  to  the  undulation  system, 
by  the  interference  of  waves  transmitted  directly,  with  those  which, 
being  reflected  at  first  at  the  second  surface  of  the  thin  plate,  are 
again  reflected  on  returning  to  the  first,  and  are  thus  sent  to  the  eye, 
at  which  they  arrive  without  any  farther  modification.  In  this  case 
the  point  where  the  surfaces  touch  should  give  a  bright  spot,  as  we 
find  by  experience  that  it  does,  so  that  here  we  have  no  additional 
principle  to  introduce  as  in  reflection  ;  but  this  is  quite  necessary  in 
many  other  cases. 

According  to  this  system,  the  thicknesses  at  which  the  rings  are 
formed  indicate  the  length  of  the  oscillations  in  any  substance.  Now 
for  one  given  mode  of  vibration  of  the  luminous  body,  the  length  of 
the  waves  must  be  equal  to  the  distance  that  the  light  passes  over 
whilst  the  vibration  takes  place ;  since,  therefore,  the  waves  are 
found  to  be  shorter  in  the  more  strongly  refracting  substances,  this 
velocity  of  transmission  must  be  less  in  them  according  to  the  same 
law ;  that  is  to  say,  it  must  be  inversely  as  the  ratio  of  refraction. 

By  considering  the  alternations  of  light  and  darkness  as  produced 
by  the  superposition  of  luminous  waves  of  the  same  or  of  a  different 
nature,  we  give  to  the  phenomenon  a  physical  chnrac'er,  and  it  is 
thus  that  Dr  Young  first  announced  the  principle  of  interne  ences ; 
but  we  may  detach  it,  as  he  has  done,  from  all  extraneous  considera- 
tions, and  present  it  as  an  experimental  law ;  it  may  then  be  ex- 
pressed as  follows ; 

(1.)  When  two  equal  portions  of  light,  in  exactly  similar  circum- 
stances, have  been  separated,  and  coincide  again  nearly  in  the  same 
direction,  they  either  are  added  together,  or  destroy  one  another,  ac- 
cording as  the  difference  of  the  times,  occupied  in  their  separate  pas- 
sages, is  an  even  or  odd  multiple  of  a  certain  half  interval  which  in 
different  for  the  different  kinds  of  light,  but  constant  for  each  kind. 

(2.)  In  the  application  of  this  law  to  different  media,  the  veloci- 
ties of  light  must  be  supposed  to  be  inversely  proportional  to  the 
ratios  of  refraction  for  those,  media,  so  that  the  rays  move  more  a  lowly 
in  the  more  strongly  refracting  medium. 


Diffraction  of  Light.  321 

(3.)  In  tfte  reflections  at  the  surface  of  a  rarer  medium,  on  some 
metals,  and  in  some  other  cases,  half  an  interval  is  lost. 

(4.)  Lastly,  it  may  be  added,  that  the  length  of  this  interval,  for 
a  given  kind  of  light,  is  exactly  four  times  that  of  the  Jits  attributed 
by  Newton  to  the  same  light. 

To  give  an  instance  of  these  laws,  suppose  that  when  two  simple 
homogeneous  rays  interfere  and  form  fringes  in  the  experiment  with 
the  two  mirrors,  you  interpose  across  the  path  of  one  of  these  a  very 
thin  plate  of  glass  that  that  ray  alone  is  to  pass  through.  According 
to  the  second  condition,  its  motion  through  the  glass  must  be  slower 
than  through  the  air  in  proportion  as  tin?  refracting  power  is  greater. 
Thus,  when  after  leaving  the  glass,  and  continuing  its  motion,  it 
meets  the  ray  with  which  it  before  interfered,  its  relations  with  this 
as  to  intervals  will  have  been  altered  ;  and  if  the  intervals  are  ever 
found  to  be  the  same,  it  must  be  when  the  ray  is  so  refracted  by  the 
glass  that  the  diminution  of  its  velocity  be  compensated  by  shorten- 
ing its  path  ;  in  this  case  the  fringes  will  be  formed  in  different 
places,  and  their  displacement  may  be  calculated  from  the  thickness 
of  the  glass  and  its  refracting  power  ;  now  this  is  confirmed  by  ex- 
periment with  incredible  exactness,  as  M.  Arago  observes,  to  whom 
we  are  indebted  for  this  ingenious  experiment. 

By  the  same  rule,  if  the  displacement  of  the  fringes  thus  produced 
by  a  given  plate  be  observed,  which  may  be  clone  with  extreme  pre- 
cision, we  may  evidently  find  the  refracting  power  of  that  plate  j 
we  may  also  compare  the  refractions  of  various  substances  by  inter- 
posing plates  of  them  successively  on  the  directions  of  the  inter- 
fering rays.  MM.  Arago  and  Fresnel  tried  this  method,  and  found 
it  so  exact  that  they  were  able  to  use  it  to  measure  differences  of 
refraction  that  no  other  method  would  have  given. 


Diffraction  of  Light. 

When  a  beam  of  light  is  introduced  into  a  dark  room,  if  you  place 
on  its  direction  the  edge  of  some  opaque  body,  and  afterwards  re- 
ceive on  a  white  surface  placed  at  a  certain  distance  that  portion  of 
the  light  which  is  not  intercepted,  the  border  of  the  shadow  will  be 
observed  to  be  edged  with  a  bright  line  ;  and  on  increasing  the  dis- 
tance, several  alternations  of  coloured  fringes  are  thus  seen  to  be 

Elem.  41 


322  Optics. 

formed.     This  phenomenon  constitutes  what  is  called  the  diffraction 
of  light. 

To  give  it  all  the  exactness  of  which  it  is  capable,  it  is  advisahle 
to  use  the  same  disposition  as  in  the  experiment  with  the  two  mir- 
rors, that  is,  to  take  a  sunbeam  directed  by  a  heliostat  and  concen- 
trated by  a  lens  almost  in  a  geometrical  point ;  an  opaque  body  is 
then  to  bp  placed  in  the  cone  of  rays  diverging  from  that  point.  To 
fix  our  ideas,  suppose  we  use  an  opaque  lamina  with  straight  edges, 
and  about  a  tenth  of  an  inch  broad  ;  if  then  the  rays  be  received  on 
a  piece  of  ground  glass  placed  at  a  certain  distance,  and  the  eye  be 
placed  beyond  this  glass,  there  will  be  observed  on  each  side  of  the 
shadow  of  the  lamina  a  numerous  series  of  brilliant  fringes  parallel 
to  the  edges,  and  separated  from  each  other  by  dark  intervals ;  the 
brightness  of  these  fringes  diminishes  as  they  recede  from  the 
shadow ;  and  the  shadow  itself  is  not  quite  dark,  but  is  formed 
also  of  luminous  and  dark  fringes  all  parallel  to  the  edges  of  the 
lamina.  Moreover,  the  ground  glass  is  not  necessary  to  exhibit 
these  fringes,  for  they  are  formed  in  the  air,  and  may  be  seen  in  it, 
either  with  the  naked  eye  or  by  the  assistance  of  a  lens  placed  ex- 
actly on  their  direction.  If  then  a  lens  be  fixed  to  a  firm  stand 
which  can  be  moved  horizontally,  by  means  of  a  screw,  along  a 
scale  divided  into  equal  parts,  its  axis  may  be  brought  successively 
opposite  each  bright  and  dark  fringe  ;  the  position  of  one  of  these 
may  be  determined  precisely  by  referring  it  to  a  fine  thread  stretch- 
ed in  front  of  the  lens,  and  thus  the  intervals  of  the  fringes  may  be 
measured,  on  the  graduated  scale,  by  the  distance  through  which  the 
lens  is  moved  to  set  it  opposite  to  each  ;  this  advantageous  arrange- 
ment was  devised  by  M.  Fresnel,  who  made  use  of  it  to  measure 
all  the  particulars  of  the  phenomenon  with  extreme  precision. 

Now  these  particulars,  as  Dr  Young  first  announced,  may  be  re- 
presented pretty  exactly  by  supposing  that  the  light  which  falls  on 
the  edges  of  the  lamina,  spreads  over  them  radiating  in  all  directions 
from  those  edges,  and  interferes  both  with  itself  and  with  the  rays 
transmitted  directly. 

The  first  kind  of  interference  forms  die  interior  fringes  ;  the  light 
radiating  from  one  edge  interfering  with  that  from  the  other,  these 
two  sets  of  rays  are  exactly  in  the  same  predicament  as  the  two 
luminous  reflected  points  in  the  experiment  of  the  mirrors  ;  thus 
also  the  disposition  of  the  interior  fringes  both  bright  and  dark,  and 
the  ratios  of  their  intervals  are  exactly  similar.  If  you  determine  in 


Diffraction  of  Light.  323 

your  mind  the  series  of  points  in  space  at  which  the  same  kind  of 
interference  takes  place  at  different  distances  behind  the  lamina, 
which  gives  the  succession  of  the  places  at  which  the  same  fringe 
appears,  you  will  find  that  those  points  are,  to  all  appearance,  on  a 
straight  line  ;  and  their  intervals,  when  measured,  are  very  exactly 
conformable  to  what  the  calculation  of  the  interferences  indicates. 

As  to  the  exterior  fringes,  they  may  be  considered  as  formed  by 
the  interference  of  the  light  transmitted  directly  with  that  radiating 
from  each  edge  ;  but  we  must  here,  as  in  the  reflected  rings,  sup- 
pose a  loss  of  an  interval  |  /.  It  thus  appears  that  the  points  at 
which  each  fringe  appears  at  different  distances  from  the  lamina,  are 
not  placed  on  a  straight  line,  but  on  a  hyperbola  of  the  second  order 
which  experiment  confirms  completely. 

We  must  not  conclude  from  this  that  diffracted  light  does  not 
move  in  straight  lines,  for  it  is  not  the  same  ray  that  forms  a  fringe 
of  a  given  order  at  different  distances.  That  the  ray  changes,  as 
the  distance  is  altered,  may  be  concluded  from  this  alone,  that  the 
fringes  may  be  observed  in  space  either  with  the  naked  eye  or  with 
a  lens ;  for  then  it  is  evident  that  the  rays  which  form  them  must 
converge,  and  afterwards  diverge  ;  otherwise  they  could  not  be  col- 
lected by  the  lens  so  as  to  afford  a  visible  image  of  their  point  of 
concourse. 

Very  remarkable  phenomena  of  diffraction  are  again  producedj 
when  the  cone  of  light,  instead  of  being  intercepted  by  an  opaque 
lamina,  is  transmitted  between  two  bodies  terminated  by  straight 
parallel  edges.  In  this  case,  the  diffracted  fringes  may,  with  great 
appearance  of  truth,  be  attributed  to  the  interference  of  the  two  por- 
tions of  light  which  fall  on  the  opposite  edges. 

Nevertheless,  there  are  many  physical  particulars  in  the  phenome- 
non, which  it  is  difficult  to  explain  on  this  hypothesis.  M.  Fresnel 
has  even  found  that  it  is  not  quite  consistent  with  the  measurements 
of  the  fringes  when  they  are  very  exact ;  he  has  been  convinced 
that  the  small  portion  of  light  which  the  edges  may  reflect  is  not 
sufficient  to  produce  the  observed  intensities  of  the  fringes ;  and  that 
it  is  necessary  to  suppose  that  other  rays  assist  which  do  not  touch 
the  edges.  He  has  thus  been  induced  to  consider  all  the  parts  of 
the  direct  luminous  wave  as  so  many  distinct  centres  of  undulations, 
the  effects  of  which  must  be  extended  spherically  to  all  the  points  of 
space  to  which  they  can  be  propagated  ;  according  to  which  suppo- 
sition, the  particular  effect  at  each  point  would  result  from  the  inter- 


324  Optics. 

ferenccs  of  all  the  partial  undulations  that  arrive  at  it.  This  consid- 
eration, applied  to  the  free  propagation  of  a  spherical  wave  in  a  ho- 
mogeneous medium,  makes  the  loss  of  light  proportional  to  the  square 
of  the  distance,  conformably  to  observation  ;  but  when  a  part  of  the 
light  is  intercepted,  it  indicates,  in  the  different  points  of  space  to- 
wards which  it  is  afterwards  propagated,  alternations  of  light  and 
darkness,  which,  in  point  of  disposition  and  intensity,  agree  most 
minutely  with  those  observed  in  diffracted  light. 

The  introduction  of  this  principle  has  enabled  M.  Fresnel  to  em- 
brace all  the  cases  of  diffraction  with  extraordinary  precision  ;  but 
an  exposition  of  his  results,  though  very  interesting,  would  lead  us 
farther  than  the  plan  of  this  work  would  allow. 


Double  Refraction. 

The  rays  of  light,  in  passing  through  most  crystallized  substances, 
are  generally  divided  into  two  parcels,  one  of  which,  containing  what 
are  called  the  ordinary  rays,  follows  the  usual  mode  of  refraction  ; 
but  the  other,  consisting  of  what  are  termed  the  extraordinary  rays, 
obeys  entirely  different  laws. 

This  phenomenon  takes  place  in  all  transparent  crystals,  except 
those  which  cleave  in  planes  parallel  to  the  sides  of  a  cube,  or  a 
regular  octaedron.  The  separation  of  the  rays  is  more  or  less 
strong,  according  to  the  nature  of  the  crystal,  and  the  direction  which 
the  light  takes  in  passing  through  it.  Of  all  known  substances,  the 
most  powerfully  double-refracting,  is  the  clear  carbonate  of  lime, 
commonly  called  Iceland  spar.  As  this  is  a  comparatively  common 
substance,  and  may  easily  be  made  the  subject  of  experiment,  we 
take  it  as  a  first  instance. 

The  crystals  of  this  variety  of  carbonate  of  lime  are  of  a  rhom- 
boidal  form,  as  represented  in  figure  100.  This  rhomboid  has  six 
acute  angles,  and  two  obtuse  ;  these  last  are  formed  by  three  equal 
plane  angles ;  in  the  acute  diedral  angles,  the  inclination  of  the 
faces  is  74°  55',  and  consequently,  in  the  others,  it  is  105°  5'.  Ma- 
lus  and  Dr.  Wollaston  have  both  found  these  values  by  the  reflec- 
tion of  light. 

If  a  rhomboid  of  this  description  be  placed  on  a  printed  book,  or 
a  paper  marked  with  black  lines,  every  thing  seen  through  it  will 
appear  to  be  double,  so  that  each  point  under  ihe  crystal  must  send 


Double  Refraction.  335 

two  images  to  the  eye,  and  consequently,  two  pencils  of  rays.  This 
indicates  that  each  simple  pencil  must  be  separated  into  two  in  its 
passage  through  the  rhomboid  ;  and  this  may  be  easily  shown  to  be  the 
case,  by  presenting  the  crystal  to  a  sunbeam,  when  it  will  give  two  dis- 
tinct emergent  beams.  To  measure  the  deviation  of  these  rays,  and 
determine  their  paths,  Malus  invented  the  following  simple  method  ; 
on  the  paper  on  which  you  place  the  rhomboid,  draw  with  very  black 
ink,  a  right-angled  triangle  ABC  (jig.  101),  of  which  let  the  least 
side  BC  be,  for  instance,  one-tenth  of  AC.  If  this  triangle  be  observ- 
ed through  the  rhomboid,  it  will  appear  double,  wherever  the  eye  be 
placed  ;  and  for  each  position  of  the  eye  there  will  be  found  a  point 
T,  where  the  line  A'C',  the  extraordinary  image  of  AC,  will  cut 
the  line  AB,  which  I  suppose  to  belong  to  the  ordinary  image.  Take 
then  on  the  triangle  itself  a  length  AF'  equal  to  A'F,  and  the  point 
F'  will  be  that  of  which  the  extraordinary  image  coincides  with  the 
ordinary  one  of  F.  The  ordinary  pencil  proceeding  from  F,  and 
the  extraordinary  one  from  .F',  are  therefore  confounded  together, 
on  emerging  from  the  crystal,  and  produce  only  one  single  pencil 
which  meets  the  eye  ;  hence,  conversely,  a  natural  pencil  proceed- 
ing from  where  the  eye  is  placed  to  the  crystal,  would  be  separated 
by  the  refraction  into  two  pencils,  one  of  which  would  go  to  F,  and 
the  other  to  F'.  This  may  indeed  be  easily  confirmed  by  experi- 
ment with  the  heliostat.  If  then  the  lines  AB,  A  C,  be  divided  each 
into  a  thousand  parts,  for  instance,  and  the  divisions  be  numbered  as 
represented  in  the  figure,  a  simple  inspection  will  suffice  to  determine 
the  points  of  AB  and  AC,  of  which  the  images  coincide  ;  conse- 
quently, if  the  position  of  these  lines  and  the  triangle  be  known,  rela- 
tively to  the  edges  of  the  base  of  the  crystal,  it  will  be  known  in  any 
case  to  what  points  of  the  base  F  and  F'  correspond,  so  that  to  con- 
struct the  refracted  rays,  it  will  only  remain  to  determine  on  the 
upper  surface,  the  position  of  their  common  point  of  emergence 
(Jig.  102).  This  might  be  done  by  marking  on  that  surface  the  point 
/,  where  the  images  of  AB  and  AC  intersect;  but  as  it  is  useful 
also  to  know  the  direction  of  the  emergent  pencil,  it  is  better  to  make 
the  observation  with  a  graduated  circle  placed  vertically  in  the  plane 
of  emergence  IOV.  The  sights  of  this  circle  must  be  directed  to 
the  point  /,  and  if  the  precaution  has  been  taken  of  levelling  the 
plane  on  which  the  crystal  lies,  the  same  observation  will  determine 
at  once  the  angle  of  emergence  IOV,  or  JV/O,  measured  from  the 
normal,  and  the  position  of  the  point  Jon  the  rhomboid.  The  posi- 


326  Optics. 

tions  of  the  points  F,  F/,  are  also  known  a  priori,  so  that  the  direc- 
tions FIj  F'l,  may  be  constructed  ;  whereupon  we  may  remark,  that 
in  many  cases  the  extraordinarily  refracted  pencil  F'l  does  not  lie 
in  the  plane  of  emergence  JV/O. 

Such  is  the  process  devised  by  Mains  ;  if  we  admit  it,  we  may 
admit  also  all  his  observations,  and  consider  them  as  data  to  be  satis- 
fied, but  I  will  shortly  indicate  a  more  simple  method,  which  would 
allow  us  to  repeat  these  same  measurements  with  equal  facility  and 
accuracy. 

Among  all  the  positions  that  may  be  given  to  the  crystal,  resting 
always  on  the  same  face,  there  is  one  which  deserves  particularly  to 
be  remarked,  because  the  extraordinary  refraction  takes  place,  like 
the  ordinary,  in  the  plane  of  emergence.  To  find  this  position,  it  is 
necessary  to  conceive  a  vertical  plane  to  pass  through  the  side  BC 
of  the  triangle,  to  place  the  eye  in  this  plane,  and  slowly  turn  the 
crystal  round  on  its  base,  till  the  two  images  of  BC  coincide  ;  then, 
as  the  ordinary  image  is  always  in  the  plane  of  emergence,  the  ex- 
traordinary must  in  that  case  be  in  it  likewise.  The  particular  plane 
for  which  this  takes  place  is  called  the  principal  section  of  the  rhom- 
boid. If  the  crystal  used  in  the  experiment  be  of  the  primitive  form, 
for  the  carbonate  of  lime,  the  bases  of  the  rhomboid  will  be  perfect 
rhombs,  and  the  principal  section  will  be  that  containing  the  shorter 
diagonals  of  the  upper  and  lower  faces.  This  section  of  the  rhom- 
boid will  be  a  parallelogram  ABA'B'  (Jig.  103),  in  which  AB, 
A'B',  are  the  diagonals  just  mentioned,  and  JIB',  A'B,  edges  of  the 
rhomboid.  The  line  AA  is  called  the  axis  of  the  crystal ;  it  is 
•equally  inclined  to  all  the  faces,  forming  with  them  angles  of  45° 
23'  25".  It  is  to  this  line  that  all  the  phenomena  of  double  refrac- 
tion are  referred. 

Let  us  examine  at  first  the  manner  of  this  refraction  in  the  princi- 
pal section.  All  its  general  phenomena  are  exhibited  in  figure  104, 
in  which  SI  represents  an  incident  ray,  IO  the  ordinary  refracted 
ray,  IE  the  extraordinary  ;  /JV  is  the  normal.  When  the  incidence 
is  perpendicular,  the  ordinary  ray  is  confounded  with  the  normal, 
and  passes  through  the  crystal  without  deviation  ;  but  the  extraordi- 
nary is  refracted  at  the  point  of  incidence,  and  is  more  or  less  de- 
flected towards  the  lesser  solid  angle  B'.  A  similar  effect  is  observ- 
ed in  every  other  case,  as  shown  in  the  figure,  the  extraordinary  ray 
lying  always  on  the  same  side  of  the  ordinary. 


Double  Refraction. 


The  inference  to  be  drawn  from  this  is,  that  there  exists  in  the 
crystal  some  peculiar  force  which  abstracts  from  the  incident  pencil 
a  part  of  its  molecules,  and  repels  them  towards  B'.  But  \\hat  is 
this  force  ?  We  shall  soon  see  that  it  emanates,  or  seems  to  emanate 
from  the  axis  of  the  crystal  ;  that  is,  that  if  through  each  point  of 
incidence  there  be  drawn  a  line  IA'  parallel  to  that  axis,  and  repre- 
senting its  position  in  the  first  strata  in  which  the  pencil  is  divided, 
all  the  phenomena  take  place  just  as  if  there  emanated  from  that  line 
a  repulsive  force,  which  acted  only  on  a  certain  number  of  luminous 
particles,  and  tended  to  drive  them  from  its  direction.  This  force 
always  throws  the  rays  towards  B',  because  they  are  always  found 
on  that  side  of  the  axis,  under  whatever  angle  of  incidence  they  may 
have  entered. 

Let  us  follow  up  this  idea,  which  does  not  appear  repugnant  to 
the  few  observations  that  have  been  made,  and  to  verify  it  by  a  direct 
experiment,  let  us  divide  the  crystal  by  two  planes  perpendicular  to 
its  axis  (Jig.  J05),  so  as  to  form  two  new  faces  abc,  a'b'c',  parallel 
to  each  other.  Now  if  we  direct  a  ray  SI  perpendicularly  to  those 
faces,  it  will  penetrate  them  in  a  direction  parallel  to  the  primitive 
axis  of  the  crystal.  Supposing  then  that  the  repulsive  force  ema- 
nates from  that  axis,  it  will  be  nothing  in  this  case,  and  the  incident 
rays  will  not  be  separated.  This  is,  in  fact,  what  takes  place  ;  there 
is  in  this  case  but  one  image. 

It  is  even  found,  in  making  the  experiment,  that  the  image  remains 
single,  when  the  second  face  of  the  plate  is  inclined  to  the  axis,  pro- 
vided that  the  first  be  perpendicular  to  it,  and  to  the  incident  rays. 
This  would  happen,  for  instance,  if  only  the  first  solid  angle  A  of 
the  primitive  rhomboid  were  taken  off.  The  incident  ray  SI  would 
continue  its  progress  parallel  to  the  axis,  as  before,  and  on  emerging 
from  the  second  surface,  it  would  be  refracted  in  one  single  direction, 
according  to  the  law  of  ordinary  refraction.  Hence,  we  may  con- 
clude, conversely,  that  an  incident  ray  R'l',  which  passed  out  of  air 
into  such  a  prism  under  the  proper  angle  of  incidence,  would  be  re- 
fracted in  one  single  ray  parallel  to  the  axis,  and  emerge  at  /  in  the 
same  manner.  This  again  is  confirmed  by  experiment.  If,  after 
having  cut  TL  rhomboid  in  the  manner  described,  the  eye  be  applied 
to  the  face  which  is  perpendicular  to  the  axis,  so  as  to  receive  only 
the  rays  which  arrive  in  that  direction,  all  the  images  of  external  ob- 
jects will  be  single  ;  they  only  undergo  at  their  edges  the  diffusion 
which  belongs"  to  the  general  phenomenon  of  the  decomposition  of 
light  by  the  unequal  refractions. 


328  Optics. 

But  if  the  repulsive  force  which  produces  the  extraordinary  re- 
fraction, really  emanates  from  the  axis,  as  the  phenomena  seem  to 
indicate,  it  cannot  disappear,  except  when  the  incident  ray  is  parallel 
to  the  axis.  The  section,  then,  which  we  have  described,  is  the 
only  one  in  which  a  crystal  prism  can  give  a  single  image ;  this  again 
is  confirmed  by  experiment,  and  we  might  avail  ourselves  of  this 
character,  to  find  the  position  of  the  axis  in  any  piece  of  Iceland 
spar,  not  in  the  primitive  form. 

To  return  to  our  plate  with  parallel  faces,  cut  perpendicularly  to 
the  axis.  We  have  seen  that  the  rays  are  not  separated  when  they 
are  incident  perpendicularly  ;  but  when  they  enter  obliquely,  they 
ought  to  be  separated,  since  they  then  form  a  certain  angle  with  the 
axis,  from  which  the  repulsive  force  emanates.  This  is  really  what 
takes  place ;  and,  moreover,  for  equal  angles  of  incidence,  the  ex- 
traordinary refraction  is  the  same  on  all  sides  of  the  axis,  which 
shows  that  the  repulsive  force  acts  from  the  axis  equally  in  all  direc- 
tions. 

Many  other  crystallized  substances,  very  different  from  the  Ice- 
land spar,  exhibit  like  it  a  certain  single  line  or  axis,  round  which 
their  double  refraction  is  exerted  symmetrically,  being  insensible  for 
rays  parallel  to  that  axis,  and  increasing  with  their  inclination  to  it,  so 
as  to  be  strongest  for  those  which  are  at  right  angles  to  the  axis. 
Crystals  thus  constituted  are  called  crystals  with  one  axis.  For  in- 
stance, quartz,  commonly  called  rock  crystal,  has  an  axis  parallel  to 
the  edges  of  the  hexaedral  prism,  under  the  form  of  which  it  is  gen- 
erally found.  But  there  is  between  its  double  refraction  and  that  of 
the  spar,  this  capital  difference,  observed  by  M.  Biot,  that  in  the 
spar  the  deviation  of  the  extraordinary  rays  from  the  axis,  is  greater 
than  that  of  the  ordinary,  whereas  in  quartz  crystals  it  is  less.  All 
crystals  with  one  axis,  that  he  has  examined,  have  been  found  to 
possess  one  or  other  of  these  modes  of  action,  which  has  occasioned 
their  distinction,  by  him,  into  crystals  of  attractive  and  repulsive  dou- 
ble refraction  ;  these  denominations,  which  express  at  once  the  phe- 
nomena, are  useful  in  innumerable  cases,  to  indicate  how  the  extra- 
ordinary ray  is  disposed  with  respect  to  the  other,  since  it  is  only 
necessary  afterwards  to  know  the  direction  of  the  axis  at  the  point 
where  the  refraction  and  separation  of  the  rays  take  place.  The 
progressive  and  increasing  separation  of  the  rays,  as  their  direction 
deviates  more  and  more  from  the  axis  in  each  of  these  classes  of 
crystals,  may  also  be  conveniently  expressed  by  saying,  that  the  phe- 


Double  Refraction.  329 

nomena  take  place  as  if  there  emanated  from  the  axis  a  force  attrac- 
tive in  the  one  class,  and  repulsive  in  the  other ;  which  does  not, 
however,  imply  a  belief  that  such  forces  do  actually  exist,  or  are  im- 
mediately exerted. 

There  are,  however,  other  crystals  in  great  number,  in  which  the 
double  refraction  disappears  in  two  distinct  directions,  forming  an 
angle  more  or  less  considerable,  so  that  rays  are  singly  refracted 
along  those  two  lines,  but  are  separated  more  and  more  widely  as 
their  incident  direction  deviates  from  them  ;  crystals  of  this  kind  have 
been  called  crystals  with  two  axea.  In  those  which  have  hitherto 
been  examined,  it  has  been  found  that  one  of  die  refractions  is 
always  of  the  ordinary  kind,  as  if  the  substance  was  not  crystallized, 
whilst  the  other  follows  a  law  analogous  to  that  of  the  crystals  with 
one  axis,  but  more  complex,  which  will  be  afterwards  explained. 
Tiiere  are  here,  as  in  the  simpler  case,  two  classes  distinguished  by 
attractive  and  repulsive  double  refraction.  No  crystals  have  as  yet 
been  discovered,  possessing  more  than  two  directions  of  single  re- 
fraction, except  indeed  those  in  which  it  is  single  in  all  directions, 
which  is  the  case  with  those  of  which  the  primitive  form  is  either  a 
cube,  or  a  regular  octaedrou.* 

The  general  circumstances  which  characterize  the  phenomenon 
of  double  refraction,  being  thus  recognised,  its  effects  must  be  ex- 
actly measured  in  each  class  of  crystals,  in  order  to  try  and  discover 
the  laws  of  it.  In  order  to  this,  there  is  no  better  plan  to  be  pursued, 
than  to  cut  them  into  plates,  or  prisms  in  different  directions,  rela- 
tively to  the  axis,  to  observe  the  extraordinary  refractions,  under 
different  incidences,  and  endeavour  to  comprise  them  in  one  general 
law.  This  Huygens  has  done  for  Iceland  spar.  The  empiric  law 
inferred  by  him,  has  been  since  verified  by  Dr  Wollaston,  and  sub- 
sequendy  by  Malus,  by  means  of  direct  experiments,  which  have 
confirmed  the  exactness  of  it.  M.  Blot  has  made  similar  experi- 
ments with  other  crystals  of  both  classes,  by  means  of  a  very  simple 
apparatus,  which  affords  very  exact  measurements  of  the  deviations 
of  the  rays,  even  in  cases  where  the  double  refraction  is  very  weak. 
As  observations  of  this  kind  are  indispensable,  as  the  foundations  of 

*  This  important  remark  of  the  connexion  between  the  primitive 
form  of  a  crystal,  and  its  single  or  double  refraction,  is  due  to  Dufay, 
who  was  likewise  the  discoverer  of  the  distinction  between  the  vitre- 
ous and  resinous  electricities. 

Elcm.  42 


330  Optics. 

all  theory,  it  will  be  as  well  to  give  here  a  detailed  description  of  the 
apparatus. 

It  consists  principally  of  two  ivory  rulers  ./2X,  AZ,  (fg-  106) 
divided  into  equal  parts,  and  fixed  at  a  right  angle.  The  former, 
jlX,  is  placed  on  a  table ;  the  other  becomes  vertical.  A  little  pillar 
H  A,  of  which  the  top  and  bottom  are  parallel  planes,  is  moveable 
along  JtX,  and  may  therefore  be  placed  at  any  required  distance 
from  AZ. 

This  disposition  is  sufficient,  when  the  extraordinary  refraction  to 
be  observed  takes  place  in  the  same  plane  as  the  ordinary,  which 
we  have  seen  to  be  the  case  under  particular  circumstances.  As 
this  is  the  simplest  case,  and  is  all  that  is  necessary  to  understand  the 
method,  I  will  explain  it  first. 

If  the  substance  to  be  observed,  had  a  very  strong  refracting 
power,  it  would  be  sufficient  to  form  a  plate  of  it  with  parallel  sur- 
faces, upon  which  experiments  might  be  made  in  the  manner  about 
to  be  described  ;  but  this  case  being  of  rare  occurrence,  we  will  sup- 
pose, in  general,  that  the  crystal  is  cut  into  a  prismatic  form,  to  make 
its  refraction  more  sensible ;  it  is  even  advisable  to  give  the  prism  a 
very  large  refracting  angle,  a  right  angle,  for  instance,  (Jig.  107)  which 
has  the  particular  advantage  of  simplifying  calculations.  As,  however, 
the  rays  of  light  cannot  pass  through  both  sides  of  such  a  prism,  of  any 
ordinary  solid  substance  placed  in  air,  being  reflected  at  the  second 
surface,  there  must  be  fixed  to  this  surface,  represented  by  CD  in 
figure  108,  another  prism,  or  parallclopiped  of  glass  CFED,  of 
which  the  refracting  angle  D,  is  nearly  equal  to  the  angle  C  of  the 
crystal  prism,  so  that  the  faces  CB,  DE,  of  the  crystal  and  glass, 
may  be  nearly  parallel.  The  two  prisms  are  to  be  joined  together, 
by  heating  them,  and  melting  between  them  a  few  grains  of  very 
pure  gum-mastic,  which  on  being  pressed,  will  spread  into  a  very 
thin  transparent  layer.  This,  when  cooled,  will  he  quite  sufficient 
to  make  the  prisms  cohere  together  very  strongly,  and  to  let  the  rays 
pass  from  one  into  the  other. 

The  double  prism  is  to  be  placed  on  the  pilkr  H h,  as  in  the  fig- 
ure, and  the  observer  is  to  look  through  it  at  the  vertical  scale  AZ. 
This  scale  will  appear  double,  the  ordinary  and  extraordinary  image 
being,  in  the  simple  case  here  considered,  in  the  same  vertical  line. 
Now  whatever  be  the  law  of  the  two  refractions,  the  corresponding 
lines  of  the  two  scales  seen,  arc  never  equally  separated  in  all  places, 
so  that  if  in  one  part  the  separation  amounts  to  half  a  degree  of  the 


Double  Refraction.  331 

scale,  a  little  further  on  it  will  be  a  whole  degree,  in  another  place  a 
degree  and  a  half,  two  degrees,  and  so  on.  If,  for  instance,  number 
451  of  the  extraordinary  division,  which  we  will  represent  by  45 le, 
coincides  with  number  450,  of  the  ordinary  (4500),  so  that  here  the 
separation  of  the  images  is  of  one  degree,  it  will  perhaps  be  found 
that  502C  falls  on  500  0.  This  shows  that  the  extraordinary  rays 
coming  from  502,  enter  the  eye  together  with  the  ordinary  from  500, 
and  since  the  glass  prism  can  produce  no  effect  beyond  simple  re- 
fraction on  these  rays,  it  is  certain  that  the  rays  from  500  0  and  502e, 
must  coincide  at  their  emergence  from  the  crystal.  This  condition 
furnishes  a  very  accurate  method  to  verify  the  law  followed  by  the 
extraordinary  rays  in  the  crystal.  In  fact,  the  directions  of  incidence 
of  the  two  pencils  may  be  determined,  since  one  of  them  El,  pro- 
ceeds from  the  point  E  of  the  scale,  of  which  the  place  is  known 
from  the  graduation,  and  arrives  at  the  point  of  incidence  /,  the  posi- 
tion of  which  is  also  determined  by  the  known  height  of  the  pillar 
Hh,  and  its  position  on  the  horizontal  scale.  There  are  similar 
data  for  the  other  O  i,  which  undergoes  only  the  ordinary  refraction, 
whether  its  point  of  incidence  be  supposed  the  same  as  that  for  EIt 
or  whether  the  small  distance  of  those  points  be  estimated  by  calcu- 
lation, taking  into  account  the  thickness  of  the  crystal  prism,  as  will 
be  hereafter  mentioned. 

Now  if  the  ordinary  refracted  pencil  Ol  be  followed  through  the 
crystal,  which  may  be  done  by  the  common  law  of  refraction,  it  may 
be  traced  to  its  emergence  from  the  second  surface  CD.  Thus  it 
will  only  remain  to  calculate  the  position  of  the  extraordinary  pencil, 
which  should  enter  the  crystal  by  the  same  surface,  accompanying 
the  exterior  ray  I" I' ;  and  following  back  this  pencil  through  the 
prism,  to  the  first  surface  by  an  assumed  law,  for  the  extraordinary 
refraction,  it  will  be  seen  whether  it  coincides,  as  it  ought,  with  the 
incident  pencil  El.  It  is  not  irrelevant  to  remark  that  this  condition 
and  indeed  every  part  of  the  observation,  is  quite  independent  of  the 
greater  or  less  refracting  power  of  the  glass  prism  CDKF,  which 
serves  merely  to  receive  the  rays  refracted  into  the  crystal,  and  make 
their  emergence  possible. 

In  the  above  instance,  I  have  supposed  the  crystal  to  be  cut  so 
that  the  extraordinary  refraction  took  place  in  the  vertical  plane,  like 
the  ordinary  ;  that  is  the  simplest  case  ;  but  when  there  is  a  lateral 
deviation,  I  place  perpendicularly  to  the  vertical  division,  a  divided 
ruler  RR  (Jig.  109),  which  is  fixed  at  the  point  from  which  the 


332  Optics. 

refracted  rays  proceed.  Then  there  are  observed  certain  lateral 
coincidences  on  the  scale  of  RR,  on  each  of  the  vertical  rods, 
if  the  direction  of  the  point  or  line  of  incidence  be  marked  on  the 
first  surface  of  the  crystal,  by  a  small  line  drawn  on  it,  or  by  means 
of  a  little  strip  of  paper  stuck  to  it,  to  limit  the  incidence  of  the  rays 
of  which  the  common  incidence  is  observed. 

Similar  means  are  used  to  fix  the  heights  of  the  points  of  inci- 
dence on  the  crystal,  when  the  coincidences  are  observed  on  the 
vertical  scale,  but  then  the  edge  of  the  strip  of  paper  must  be  put 
horizontal. 

One  may  even  observe  the  coincidences  on  the  horizontal  scale 
AX,  on  which  the  pillar  stands.  Then  the  places  of  incidence  on 
the  crystal  must  be  limited  as  before. 

One  of  the  data  of  the  calculation  must  be  the  ordinary  refracting 
power  of  the  crystal.  This  may  be  measured  by  observing  on  what 
line  of  the  horizontal,  or  vertical  scale  another  line  falls,  which  is  ob- 
served by  ordinary  refraction  through  the  double  prism,  or  through  a 
crystal  prism  of  a  smaller  angle,  without  a  glass  one.  One  may  even 
see  whether  the  ordinary  refraction  follows,  in  all  cases,  the  law  of 
the  proportionality  of  the  sines. 

It  is  necessary  to  make  the  edge  of  the  crystal  prism  as  sharp  as 
possible,  in  order  that  the  corrections  made  for  its  thickness  be  incon- 
siderable. In  fact,  the  best  way  of  making  the  observation,  when  it 
can  be  done,  is  to  let  the  rays  pass  actually  through  the  edge,  for 
then  the  two  refracted  pencils  have  but  an  infinitely  small  space  to 
pass  through,  before  they  emerge  together.  For  a  similar  reason, 
the  pillar  should,  in  the  experiments,  not  be  placed  very  near  the 
vertical  scale,  on  which  the  coincidences  are  observed,  because  the 
corrections  for  thickness,  which  are  nearly  insensible  at  moderate 
distances,  might  become  more  considerable. 

Besides  these  precautions,  the  faces  of  the  prisms  should  be 
ground  very  smooth  and  plane,  and  their  inclinations  should  be  ac- 
curately determined,  by  the  reflecting  goniometer.  Moreover,  it  is 
necessary  that  the  direction  in  which  the  prisrn  is  cut,  relatively  to 
the  axis  or  axes  of  the  crystal,  should  be  accurately  known  ;  in 
order  to  which  these  axes  should  be  previously  determined,  either 
by  immediate  observation  of  the  directions  in  which  the  reflection  is 
single,  or  by  inferences  drawn  from  the  experiments  themselves,  or 
by  other  processes  that  will  be  hereafter  detailed.  By  following 
these  rules,  the  observer  will  be,  I  believe,  perfectly  satisfied  as  to 


Double  Refraction.  333 

the  nicety  and  accuracy  of  the  mode  of  experiment.  These  advan- 
tages are  derived  from  the  multiplicity  of  the  coincidences,  seen  on 
the  doubly-refracted  scale.  The  alternate  superpositions  and  sepa- 
rations of  the  lines  of  division  produce,  if  I  may  so  express  myself, 
the  effect  of  verniers,  and  enable  one  to  judge  with  extreme  precis- 
ion, of  the  point  where  the  coincidence  is  most  perfect. 

Suppose  then,  that  by  this,  or  some  analogous  process,  we  have 
determined  for  some  given  crystal,  the  deviation  of  the  rays  in  differ- 
ent directions  round  the  axis,  it  remains  to  find  out  the  general  law, 
which  regulates  the  phenomenon  in  all  cases.  This  Huygens  has 
done,  as  has  been  before  mentioned,  for  crystals  with  one  single  axis, 
by  means  of  a  remarkable  law  that  he  connected  with  the  system  of 
undulations ;  but  this  same  law  has  since  been  deduced  by  M.  La- 
place, from  the  principle  of  material  attraction. 

If  light  is  to  be  considered  as  a  material  substance,  the  refraction 
of  its  rays  must  be  produced  by  attractive  forces,  exerted  by  the  par- 
ticles of  other  bodies  on  the  luminous  molecules,  forces  which  can 
be  sensible  only  at  very  minute  distances,  and  which  are  therefore 
quite  analogous  to  those  which  are  exerted  in  chemical  affinities. 
It  follows,  that  when  particles  of  light  are  at  a  sensible  distance  from 
a  refracting  body,  the  effect  they  experience,  from  it  is  quite  inap- 
preciable, so  that  their  natural  rectilinear  direction  is  not  altered; 
they  begin  to  deviate  from  this  direction  only  at  the  moment  when 
they  are  in  the  immediate  vicinity  of  the  refracting  surface,  and  the 
action  takes  place  only  for  an  infinitely  short  period  of  time ;  for  as 
soon  as  the  particles  have  penetrated  within  the  surface  to  a  distance 
ever  so  small,  the  forces  exerted  on  them  by  the  molecules  of  the 
medium  become  sensibly  equal  in  all  directions,  so  that  the  path  of 
the  light  becomes  again  a  straight  line,  though  different  from  the  pre- 
ceding. It  is,  therefore,  clear  that  the  curved  portion  of  the  path 
being  infinitely  small,  it  must  appear  to  consist,  on  the  whole,  of  two 
straight  lines  forming  an  angle,  which  in  fact,  is  conformable  to 
experience.  But  for  the  very  reason  that  the  curve  is  not  percepti- 
ble, it  is  useless  to  seek,  from  experiment,  any  notions  of  its  form 
that  might  lead  to  a  knowledge  of  the  laws  which  produce  it,  as  ob- 
servations on  the  orbits  of  the  planets  have  led  to  a  knowledge  of  the 
laws  of  gravitation.  We  must,  therefore,  have  recourse  to  some 
other  characters  derived  from  experiment. 

Newton  has  succeeded  in  the  case  of  ordinary  refraction,  by  con- 
sidering each  luminous  molecule  passing  through  a  refracting  surface, 


334  Optics. 

as  acted  on  before,  during,  and  after  its  passage,  by  attractive  forces 
sensible  only  at  very  small  distances,  and  emanating  from  all  parts 
of  the  refracting  medium.  This  definition  specifies  nothing  as  to  the 
law  of  the  attracting  forces  ;  it  allows  us  only  to  calculate  their  re- 
sultant for  any  distance,  and  to  suppose  that  they  become  evanescent 
when  the  distance  is  of  sensible  magnitude.  Now  these  data  are 
sufficient  to  calculate,  not  indeed  the  velocity  of  the  molecules  in 
their  curvilinear  motion,  nor  the  nature  of  that  motion,  but  only  the 
relations  of  the  final  velocities  and  directions,  which  ensue,  either  in 
the  medium  or  out  of  it,  when  the  distance  of  the  luminous  mole- 
cules from  the  refracting  surface  is  become  so  considerable  that  the 
trajectory  is  sensibly  rectilinear,  which  will  comprehend  all  distances 
that  we  can  observe. 

For  extraordinary  refraction,  we  have  not  the  advantage  of  being 
able  to  define  the  origin  of  the  molecular  force,  nor  the  manner  in 
which  it  emanates  individually  from  each  particle  of  the  crystal  j  for 
what  we  have  said  about  accounting  for  the  phenomena  by  the  sup- 
position of  attractive  and  repulsive  forces,  emanating  from  the  axes, 
is  only  the  indication  of  a  complicated  result,  and  not  the  expression 
of  a  molecular  action.  What  is  known  then,  in  this  case,  or  at  least 
what  may  be  supposed,  when  the  idea  of  the  materiality  of  light  is 
adopted,  is  that  the  forces,  whatever  they  may  be,  which  act  on  the 
rays  of  light,  in  these  as  in  other  circumstances,  are  attractive  or  re- 
pulsive, or  both,  and  emanate  from  the  axes  of  the  crystal.  Now  in 
all  cases  when  a  material  particle  is  subjected  to  the  action  of  such 
forces,  its  motion  is  subjected  to  a  general  mechanical  condition  call- 
ed the  principle  of  least  action.  Applying  this  principle  here,  and 
joining  the  particular  condition  that  the  forces  are  sensible  only  at 
insensible  distances,  M.  Laplace  has  deduced  two  equations  which 
determine  completely  and  generally  the  direction  of  the  refracted 
ray  for  each  given  direction  of  incidence,  when  you  know  the  law  of 
the  final  velocity  of  the  luminous  molecules  in  the  interior  of  the 
medium,  at  a  sensible  distance  from  the  refracting  surface. 

In  the  case  of  ordinary  refraction  the  final  velocity  is  constant,  for 
the  deviation  of  the  ordinary  ray  is  the  same  in  a  given  substance  in 
whatever  direction  the  experiment  be  made,  provided  the  angle  of 
incidence  and  the  nature  of  the  ambient  medium  be  unchanged. 
Accordingly,  if  the  interior  velocity  is  supposed  to  be  constant,  the 
equations  deduced  from  the  principle  of  least  action,  show  that  the 
refraction  takes  place  in  the  same  plane  as  the  incidence,  and 


Double  Refraction.  355 

that  the  ratio  of  the  sines  is  invariable,  as  it  appears  to  be  from  all 
observations  hitherto  made. 

Reasoning  from  analogy,  it  appeared  natural  to  suppose  that  the  ex- 
traordinary refraction  was  produced  by  a  velocity  varying  according 
to  the  inclination  of  the  ray  to  the  axes  of  the  crystal.  Now  taking' 
at  first  crystals  with  one  axis,  we  have  seen  that  the  extraordinary 
refraction  takes  place  symmetrically  all  round  the  axis,  that  it  disap- 
pears when  a  ray  lies  along  the  axis,  and  is  at  its  maximum  when 
they  are  at  right  angles.  We  must  then,  in  the  case  of  these  crys- 
tals, limit  ourselves  to  the  laws  of  velocity  that  satisfy  these  condi- 
tions. M.  Laplace  has  tried  the  following  ; 


where  v  represents  the  ordinary  velocity,  V  the  extraordinary,  &  the 
angle  between  the  extraordinary  ray  and  the  axis,  and  JTis  a  coeffi- 
cient which  is  constant  for  any  one  given  crystal.  Introducing  this 
law  of  the  velocity  in  the  equations  of  the  principle  of  least  action, 
he  obtained  immediately  Huygens'  law.  This  law  had  been  com- 
pletely verified  only  for  Iceland  spar,  but  M.  Biot  has  found  it  true 
for  quartz  and  beryl  ;  only  the  coefficient  .fiTis  positive  in  crystals  of 
attractive  double  refraction,  and  negative  in  the  others.  Its  absolute 
value  is  different  in  different  substances,  and  it  is  even  found  to  vary 
in  specimens  of  the  same  mineralogical  species;  but  with  these 
modifications  it  is  probable  that  the  law  applies  equally  to  all  crystals 
with  one  axis. 

As  to  those  having  two  axes,  it  is  clear  that  the  extraordinary 
velocity  V  must  depend  on  the  two  angles  #  and  #',  made  by  the 
refracted  ray  with  the  two  axes.  Analogy  leads  us  to  try  whether 
the  square  of  the  velocity  V  cannot  be  expressed  here  also  by  a 
function  of  the  second  degree,  but  more  general,  that  is,  depending 
on  both  the  angles;  now  in  such  crystals  the  refractions  become 
equal  when  the  ray  coincides  with  one  or  the  other  axis.  This 
proves  that  the  extraordinary  velocity  must  then  be  equal  to  the 
ordinary.  This  condition  limits  the  generality  of  the  function,  and 
reduces  it  to  the  following  form  ; 

F3  =  v2  -f-  -fiT  sin  #sin#'. 

that  is,  there  must  remain  only  the  product  of  the  two  sines.  Intro- 
ducing this  formula  into  the  equations  of  the  principle  of  least  action, 
the  path  and  motion  of  the  rays  is  found  for  all  cases,  and  it  remains 
only  to  try  wuether  it  is  conformable  to  experiment.  M.  Biot  has 


336  Optics. 

done  this  for  the  white  topaz  which  has  two  axes  of  double  refrac- 
tion, and  the  formula  agreed  perfectly  with  observation.  One  may, 
besides,  judging  by  other  phenomena  that  will  be  hereafter  indicated, 
be  convinced  that  the  same  law  applies  to  other  crystals  with  two 
axes  on  which  experiments  have  not  been  made ;  and  it  is  highly 
probably  that  it  is  universally  applicable. 

It  may  be  remarked  that  the  general  law  comprises  Huygens'  as 
a  particular  case,  for  crystals  with  only  one  axis,  considering  these  as 
having  two  axes  which  coincide,  for  then  #  and  &  become  equal, 
and  the  equation  for  V  contains  the  square  of  sin  #. 

It  will  be  seen  farther  on,  that  the  same  analogy  extends  also  to 
another  species  of  action  that  crystallized  substances  exert  on  light, 
which  will  be  explained  in  the  following  article. 


Polarisation  of  Light. 

The  polarisation  of  light  is  a  property  discovered  by  Malus,  which 
consists  in  certain  affections  that  the  rays  of  light  assume  on  buing 
reflected  by  polished  surfaces,  or  refracted  by  these  same  surfaces, 
or  transmitted  through  substances  possessing  double  refraction. 

Though  it  would  be  impossible  here  to  give  a  complete  exposition 
of  the  details  of  these  phenomena,  we  will  at  least  describe  some  of 
the  experiments  by  which  they  may  be  exhibited. 

The  first  and  principal  of  these  consists  in  giving  to  light  a  modifi- 
cation, such  that  the  rays  composing  a  pencil  will  all  escape  reflection 
when  they  fall  on  a  reflecting  surface  under  certain  circumstances. 

As  an  instance,  suppose  a  beam  of  sun-light  SI  (fig.  110)  falls 
on  the  first  surface  LL  of  a  plate  of  glass,  smooth  but  not  silvered, 
making  with  the  surface  an  angle  of  35°  25' ;  it  will  be  reflected  in 
the  direction  77',  making  the  angle  of  reflection  equal  to  that  of  inci- 
dence. Let  it  then  be  received  on  another  plate  of  glass,  smooth 
but  unsilvered,  like  the  former ;  generally  speaking,  it  will  be  again 
reflected  with  a  partial  loss.  But  the  reflection  will  cease  altogether 
if  the  second  glass  be  placed  like  the  first,  at  an  angle  of  35°  25'  to 
the  line  //',  provided  also  it  be  so  turned  that  the  second  reflection 
lake  place  in  a  plane  11' L'  perpendicular  to  that  of  the  first,  SIL. 

In  order  to  make  this  disposition  of  the  glasses  more  clearly  intel- 
ligible, we  may  imagine  that  //'  is  a  vertical  line,  thai  IS  lies  north 
and  south,  and  I'L  east  and  west. 


Polarisation  of  Light.  337 

Before  we  enter  upon  the  inferences  to  be  drawn  from  this  re- 
markable experiment,  I  will  make  a  few  observations  on  the  manner 
of  performing  it  conveniently  and  accurately. 

Many  kinds  of  apparatus  may  be  devised  to  attain  this  end.  That 
which  M.  Biot  usually  employs,  is  represented  in  figure  111.  It  is 
very  simple,  and  is  sufficient  for  all  experiments  on  polarisation.  It 
consists  of  a  tube  TT',  to  the  ends  of  which  are  fixed  two  collars 
which  turn  with  sufficient  friction  to  keep  them  fast  in  any  position. 
Each  of  them  bears  a  circular  division  which  marks  degrees.  From 
two  opposite  points  of  their  circumference  proceed  two  brass  stems 
T/7,  T'J7',  parallel  to  the  axis  of  the  tube,  and  between  them  is 
suspended  a  brass  ring  AA,  which  can  turn  about  an  axis  XX  per- 
pendicular to  the  common  direction  of  the  stems.  The  motion  of 
the  ring  is  likewise  measured  by  a  circular  graduation,  and  it  may 
be  confined  in  any  position  by  screws.  When  a  plate  of  glass  is  to 
be  exposed  to  the  light,  it  must  be  fixed  on  the  surface  of  the  ring ; 
then  it  may  be  placed  in  any  situation  whatever  with  respect  to  the 
rays  of  light  which  pass  through  the  tube  ;  for  the  collar,  turning  cir- 
cularly round  the  tube,  brings  the  reflecting  plane  into  all  possible 
directions,  preserving  a  constant  inclination  to  the  axis,  and  this  incli- 
nation may  be  varied  by  means  of  the  proper  motion  of  the  ring 
round  its  axis  XX.  The  graduated  circle  which  regulates  this  mo- 
tion should  mark  zero  when  the  plane  of  the  ring  is  perpendicular  to 
the  axis  of  the  tube,  and  the  divisions  on  the  two  collars  should  have 
their  zeros  on  the  same  straight  line  parallel  to  the  axis.  In  con- 
structing the  apparatus  one  should  take  care  that  these  conditions  are 
fulfilled  ;  but  it  is  of  no  great  consequence  that  they  be  so  exactly, 
as  any  error  may  be  compensated  by  repeating  each  observation  on 
both  sides  of  the  axis,  and  taking  the  mean  of  the  numbers  of  de- 
grees found  in  the  two  opposite  positions. 

If  it  be  desired,  for  instance,  to  repeat  Malus's  experiment  de- 
scribed above,  a  plate  of  glass  must  be  placed  on  each  ring,  and  they 
must  be  disposed  so  as  to  be  inclined  to  the  axis  at  angles  of  35°  25'. 
Then  the  graduated  circle  of  one  of  the  collars  must  be  brought  to 
mark  zero,  and  the  other  90°,  that  the  planes  of  reflection  may  be 
perpendicular  to  each  other.  The  tube  must  then  be  secured,  and 
a  candle  placed  at  some  distance  in  such  a  position  that  its  rays  may 
be  reflected  by  the  glass  along  the  axis  TT.  This  will  happen 
when  on  looking  through  the  tube  the  reflection  of  the  candle  is  seen 
in  the  first  glass.  Every  thing  being  thus  arranged,  the  reflected 
Elem.  43 


338  Optics. 

rays  will  meet  die  second  glass  at  the  same  angle  of  35°  25' ;  then 
according  to  the  different  positions  given  to  the  collar  TT  which 
carries  this  glass,  the  light  proceeding  from  the  second  reflection  will 
be  more  or  less  intense,  and  there  will  be  two  particular  positions  in 
which  there  will  be  no  rays  reflected  at  all,  of  those  at  least  which 
are  regularly  reflected  by  the  first  glass.  Care  must  be  taken  to  put 
a  dark  object  behind  the  glass  L'L'  on  the  side  opposite  to  the  re- 
flected light,  in  order  to  intercept  the  extraneous  rays  which  might 
be  sent  on  this  side  from  exterior  objects,  and  which,  passing  through 
the  glass,  and  arriving  at  the  eye,  would  mix  with  the  reflected  rays 
thai  are  the  subject  of  the  observation.  The  same  precaution  should 
be  taken  for  the  glass  LL ;  and,  indeed,  as  this  is  never  used  except 
to  reflect  light  at  its  first  surface,  the  back  of  it  may  be  blackened 
once  for  all  with  Indian  ink,  or  smoked  over  a  lamp ;  it  would  not 
do  to  silver  it  for  a  reason  that  will  be  given  hereafter. 

For  the  light  of  the  candle  mentioned  above,  may  be  substituted 
that  coming  from  the  atmosphere,  which  may  be  received  into  the 
tube  when  reflected  by  the  first  glass  LL ;  but  in  this  case  to  pre- 
serve to  the  rays  the  precise  inclination  required  for  the  phenome- 
non, the  field  of  the  tube  should  be  limited  by  some  diaphragms,  with 
very  small  apertures,  placed  within  it.  The  first  glass  should  be 
blackened  or  smoked,  as  before  mentioned,  to  intercept  any  rays  that 
might  come  by  refraction  from  objects  situated  under  it.  In  this 
manner,  on  looking  through  the  tube,  when  the  glass  LL  is  turned 
towards  the  sky,  a  small  brilliant  white  speck  will  be  seen,  on  which 
all  the  experiments  may  be  made.  The  perfect  whiteness  of  this 
spot  is  a  great  advantage ;  it  is  an  indispensable  qualification  in  many 
cases,  where  different  tints  are  to  be  observed  and  compared  ;  it  is 
impossible  to  succeed  so  well  with  the  flame  of  a  candle  or  any  other 
inflamed  substance,  as  none  of  these  flames  are  perfectly  white. 
Lastly,  the  brightness  of  the  incident  light  must  be  modified,  so  that 
the  portion  irregularly  reflected  by  the  two  glasses  may  not  be  sensi- 
ble ;  for  this  portion,  being  after  such  reflection  in  the  state  of  radi- 
ant light,  cannot  be  polarised  in  one  single  direction ;  the  other  part 
which  is  regularly  reflected,  alone  undergoes  polarisation,  and  there- 
fore alone  escapes  reflection  at  the  second  glass. 

Whatever  be  the  nature  of  the  apparatus  employed,  the  process 
will  always  be  the  same,  and  the  same  phenomena  of  reflection  will 
be  observed  on  the  second  glasss.  To  exhibit  them  in  a  methodical 
manner,  which  will  allow  us  easily  to  take  them  all  in  at  one  view, 


Polarisation  of  Light.  339 

we  will  suppose,  as  above,  that  S/L,  the  plane  of  incidence  of  the 
light  on  the  first  glass,  coincides  with  that  of  the  meridian,  and  that 
the  reflected  ray  If  is  vertical.  Then  if  the  collar  TT  which 
bears  the  second  glass  be  turned  round,  this  glass  will  also  turn  all 
round  the  reflected  ray,  making  always  the  same  angle  with  it,  and 
the  second  reflection  will  be  directed  successively  to  all  the  different 
points  of  the  horizon  ;  this  being  premised,  the  phenomena  that  will 
be  observed  are  as  follows ; 

When  the  second  or  lower  glass  is  placed  so  that  the  second  re- 
flection takes  place  in  the  plane  of  the  meridian  like  the  first,  the 
intensity  of  the  light  finally  reflected  is  at  its  maximum.  As  this 
glass  is  turned  round,  it  reflects  less  and  less  of  the  light  thrown  on  it. 

Finally,  when  the  lower  glass  faces  the  east  or  west  point,  the 
light  passes  altogether  through  it  without  being  reflected  at  either 
surface. 

If  the  collar  be  turned  still  farther  round,  the  same  phenomena 
recur  in  an  inverse  order,  that  is,  the  intensity  of  the  light  reflected 
increase,  by  the  same  degrees  as  it  diminished  before,  and  attains  the 
same  maximum  state  when  directed  towards  the  meridian,  and  so  on 
through  the  whole  circle. 

It  appears  then,  that  during  a  whole  revolution  of  the  glass  the 
intensity  of  the  reflected  light  has  two  maxima  answering  to  the  azi- 
muths 0  and  180°,  and  two  minima  answering  to  90°  and  270°. 
Moreover,  the  variations  are  quite  similar  on  different  sides  of  these 
positions.  These  conditions  will  be  completely  satisfied  by  suppos- 
ing, as  Mains  does,  that  the  intensity  varies  as  the  square  of  the  co- 
sine of  the  angle  between  the  first  and  second  planes  of  reflection. 

The  results  of  this  interesting  observation  being  thus  collected  into 
one  point  of  view,  we  may  draw  this  general  consequence  from  them, 
that  a  ray  reflected  by  the  first  surface  is  not  reflected  by  the  second, 
(under  a  particular  incidence)  when  it  presents  its  east  or  west  side 
to  the  surface,  but  that  in  all  other  positions  it  is  more  or  less  re- 
flected. Now  if  light  be  a  matter  emitted,  a  ray  of  light  can  be 
nothing  else  but  the  rapid  succession  of  a  series  of  molecules,  and 
the  sides  of  it  are  only  the  different  sides  of  these  molecules.  We 
must,  therefore,  necessarily  conclude  that  these  have  faces  endowed 
with  different  physical  properties,  and  that  in  the  present  case  the 
first  reflection  turns  towards  the  same  point  of  space,  faces,  if  not 
similar,  at  least  endowed  with  similar  properties.  This  arrangement 
of  the  molecules  Malus  denominated  the  polarisation  of  light,  assimi- 


340  Optics. 

lating  the  operation  of  the  first  glass  to  that  of  a  magnet  which  turns 
the  poles  of  a  number  of  needles  all  in  the  same  direction. 

Hitherto  we  have  supposed  that  the  incident  and  reflected  rays 
made  angles  of  35°  25'  with  the  glasses ;  it  is  indeed  only  under  that 
angle  that  the  phenomenon  takes  place  completely.  If  while  the 
first  glass  remains  fixed,  the  inclination  ol  the  second  to  the  ray  be 
ever  so  little  altered,  it  will  be  found  that  the  second  reflection  will 
not  be  entirely  destroyed  in  any  position,  though  it  will  still  be  at  a 
minimum  in  the  east  and  west  plane.  If  again,  the  inclination  of  the 
ray  to  the  second  glass  being  preserved,  that  on  the  first  be  changed, 
it  will  be  seen  that  the  ray  will  never  pass  entirely  through  the  sec- 
ond glass,  but  the  partial  reflections  which  take  place  at  its  surfaces 
are  at  a  minimum  in  the  above  mentioned  position. 

Similar  phenomena  may  be  produced  by  means  of  most  transpa- 
rent substances  besides  glass.  The  two  planes  of  reflection  must 
always  be  at  right  angles,  but  the  angle  of  incidence  varies  with  the 
substance.  According  as  the  refracting  power  of  this  is  greater  or 
less  than  that  of  the  ambient  medium,  the  angle  of  polarisation,  mea- 
sured from  the  surface,  is  greater  or  less  than  half  a  right  angle. 
We  have  seen  that  for  glass  this  angle  is  35°  25' ;  for  sulphate  of  ba- 
rytes  it  is  only  32°,  and  for  diamond  only  23°.  If  glass  plates  be 
placed  in  essential  oil  of  turpentine  which  has  a  refracting  power 
almost  exactly  equal  to  that  of  glass,  the  angle  of  polarisation  will  be 
found  to  differ  very  little  indeed  from  45°.  The  reflection  at  the 
second  surface  is  supposed  to  take  place  on  the  ambient  medium 
which  bounds  the  glass.  In  general,  according  to  an  ingenious  re- 
mark of  Dr  Brewster's,  the  angle  of  polarisation  is  characterized  by 
the  reflected  ray  being  perpendicular  to  the  refracted.  The  angles 
calculated  on  this  hypothesis  agree  singularly  well  with  experiment, 
and  also  confirm  the  rule  given  above  for  the  different  magnitudes 
of  them,  as  will  easily  appear  from  figures  112,  113,  and  114,  in 
which  the  refracting  power  is  supposed  to  be  respectively  greater 
than  unity,  equal  to  that  number,  and  less  than  it. 

This  law  applies  equally  well  to  substances  which,  like  the  dia- 
mond and  sulphur,  never  produce  more  than  an  incomplete  polarisa- 
tion, for  the  quantity  of  light  reflected  is.  invariably  a  minimum  for 
the  angle  so  determined. 

If  the  mode  of  observation  which  we  have  applied  to  smooth  glass 
plates  be  universally  employed,  it  may  serve  to  show  that  polarisa- 
tion when  complete  is  always  a  modification  exactly  of  the  same 


Polarisation  of  Light.  341 

kind  for  all  substances ;  for  when  a  beam  of  light  has  been  once 
polarised,  it  will  equally  pass  through  all  substances,  with  the  excep- 
tion mentioned  above,  provided  each  be  presented  to  it  under  its 
proper  angle  ;  and  whatever  be  the  nature  of  the  first  or  second  sub- 
stance employed,  'the  variation  of  intensity  in  the  light  after  the  sec- 
ond reflection  is  always  subject  to  the  same  laws. 

To  represent  these  circumstances  geometrically,  let  us  consider  a 
a  ray  //'  (flg.  115)  polarised  by  reflection  on  a  glass  plate  LL,  and 
through  any  one  of  the  molecules  composing  it,  let  there  be  drawn 
three  rectangular  axes  c  z,  ex,  cy,  the  first  coinciding  with  the  ray, 
the  second  in  the  plane  of  reflection  SIC,  the  third  perpendicular  to 
both  the  others.  Then  when  the  ray  IF  meets  a  second  glass  L'L' 
placed  so  as  to  produce  no  reflection,  the  reflecting  forces  which  em- 
anate perpendicularly  from  the  glass,  must  be  perpendicular  to  the 
axis  ex ;  moreover  they  must  act  equally  on  molecules  lying  towards 
c  x,  and  towards  c  a/,  for  if  the  glass  be  turned  a  little  from  the  posi- 
tion of  no  reflection,  the  effects  are  found  to  be  symmetrical  on  all 
sides  of  that  position.  The  action,  therefore,  of  these  reflecting 
forces,  in  this  position,  cannot  make  the  axis  xcx/  turn  either  to  the 
right  or  left,  any  more  than  the  force  of  gravity  can  turn  a  horizontal 
lever  with'  equal  arms.  They  cannot  bring  the  axis  into  their  own 
plane,  in  which  we  see  it  was  in  the  first  reflection,  by  which  the 
polarisation  took  place  on  the  glass  LL.  This  proves  that  it  is  on 
that  axis  that  the  properties  of  the  luminous  molecules  depend.  We 
will  for  that  reason  call  it  the  axis  of  polarisation,  and  suppose  its 
direction  similar  and  invariably  determined  for  each  molecule.  Far- 
ther, for  the  sake  of  conciseness,  we  will  call  c  z  the  axis  of  transla- 
tion ;  but  we  do  not  suppose  this  invariable  in  each  molecule,  and 
we  will  consider  it  only  as  relative  to  its  actual  direction,  in  order  to 
leave  each  molecule  at  liberty  to  turn  round  its  axis  of  polarisation. 
According  to  these  definitions  all  the  results  that  we  have  hitherto 
obtained  may  be  enounced  very  simply  and  clearly  in  the  following 
manner ; 

When  a  ray  of  light  is  reflected  by  a  polished  surface,  under  the 
angle  ivhich  produces  complete  polarisation,  the  axis  of  polarisation 
of  every  reflected  molecule  is  situated  in  the  plane  of  reflection,  and 
perpendicular  to  the  actual  axis  of  translation  of  that  molecule. 

If  the  incident  molecules  are  turned  so  that  this  condition  cannot 
possibly  be  fulfilled,  they  will  not  be  reflected  at  least  under  the  an- 
gle of  complete  polarisation.  This  happens  when  the  axis  of  polari- 


342  Optic*. 

sation  of  an  incident  molecule  is  perpendicular  to  the  plane  of  inci- 
dence, the  angle  of  incidence  being  properly  determined  a  priori. 

Generally  speaking,  when  a  polished  surface  receives  a  polarised 
ray  under  the  angle  at  which  it  would  itself  produce  polarisation,  if 
it  be  made  to  turn  round  the  ray  without  changing  that  angle,  the 
quantity  of  light  reflected  in  different  positions  varies  as  the  square 
of  the  cosine  of  the  angle  between  the  plane  of  incidence,  and  die 
axis  of  polarisation. 

When  a  ray  of  light  has  undergone  polarisation  in  a  certain  direc- 
tion, by  the  process  above  described,  it  carries  that  property  with  it, 
and  preserves  it  without  sensible  alteration,  when  made  to  pass  per- 
pendicularly through  even  considerable  thicknesses  of  air,  water,  and 
in  general,  any  substance  that  »  xerts  only  single  refraction ;  but  dou- 
ble-refracting media  alter,  in  general,  the  polarisation  of  a  ray,  and 
in  a  manner,  to  all  appearance,  sudden,  communicating  to  it  a  new 
polarisation  of  the  same  nature  in  a  different  direction.  It  is  only 
when  crystals  are  held  in  certain  directions,  that  the  ray  can  escape 
this  disturbing  influence.  Let  us  endeavour  to  compare  more  closely 
these  two  kinds  of  action. 

That  of  single-axed  crystals  has  been  studied  by  Malus,  who  has 
comprised  its  effects  in  the  following  law.  When  a  pencil  of  light 
naturally  emanating  from  a  luminous  body,  passes  through  a  single- 
axed  crystal,  and  is  divided  into  two  pencils  having  different  direc- 
tions, each  of  these  pencils  is  polarised  in  one  single  direction  ;  the  or- 
dinary one  in  the  plane  passing  through  its  direction  and  a  line  parallel 
to  the  axis  of  the  crystal,  the  extraordinary  one  perpendicularly  to  a 
plane  similarly  situated  with  respect  to  its  direction.  Either  of  these 
rays,  when  received  on  a  plate  of  glass  after  its  emergence,  shows 
all  the  characters  of  polarisation  that  we  have  described. 

This  law  subsists  equally,  when  the  ray  has  been  polarised  by  re- 
flection before  its  passage  through  the  crystal.  The  two  refracted 
pencils  are  always  polarised,  as  if  they  had  been  composed  of  direct 
rays,  but  their  relative  intensities  differ  according  to  the  direction  of 
the  primitive  polarisation  given  to  them  ;  this  direction  must  there- 
fore have  predisposed  the  particles  to  undergo  in  preference  one  or 
other  of  the  refractions. 

These  two  laws  were  discovered  by  Malus.  The  analogy  remarked 
above  between  the  single-axed  and  double-axed  crystals,  indicates 
sufficiently  how  it  is  to  be  extended  to  the  latter  ;  to  find  the  direc- 
tion of  polarisation  for  the  ordinary  pencil,  draw  a  plane  through  its 


Polarisation  of  Light.  343 

direction,  and  through  each  of  the  axes  of  the  crystal.  If  either  of 
these  axes  existed  alone,  the  ordinary  pencil  would  be  polarised  in 
the  plane  belonging  to  it.  Now  it  is  really  found  polarised  in  a  plane 
intermediate  to  those  two,  and  the  extraordinary  pencil  perpendicu- 
larly to  the  analogous  plane  drawn  through  its  direction  between 
the  two  planes  containing  the  axes.  If  the  angle  between  these  be 
equal  to  nothing,  the  crystal  is  single-axed,  and  the  direction  of  po- 
larisation is  conformable  to  Malus's  indications  ;  this  law  has  been 
directly  verified  on  the  two  pencils  refracted  by  the  topaz ;  as  for 
other  crystals  in  which  it  has  not  been  possible  to  verify  it  directly, 
we  may,  by  the  consideration  of  some  other  phenomena  that  will 
shortly  be  mentioned,  judge  that  it  applies  to  them  also. 

These  laws  of  polarisation  are  applicable  in  all  cases  where  the 
two  pencils  transmitted  by  a  crystal  are  observed  separately,  but 
when  they  are  received  simultaneously,  and  in  nearly  the  same  di- 
rection, that  of  their  apparent  polarisation  is  found  to  be  modified, 
and  at  the  same  time  their  coincidence  produces  certain  colours, 
which  M.  Arago  first  observed,  and  of  which  M.  Biot  determined 
the  experimental  laws.  The  most  simple  arrangement  to  exhibit 
these  colours,  is  to  place  a  thin  lamina  of  some  crystallized  substance, 
in  the  direction  of  a  white  ray,  previously  polarised  by  reflection,  and 
to  analyze  the  transmitted  light  by  means  of  a  double-refracting 
prism.  The  light  is  thus  separated  into  two  portions,  of  which  the 
colours  are  complementary  to  each  other,  and  identical  with  those  of 
the  rings  between  two  glasses.  One  of  these  portions  appears  to 
have  preserved  its  primitive  polarisation,  whilst  the  other  exhibits  a 
new  polarisation,  of  which  the  direction  depends  on  that  given  to  the 
axes  of  the  crystal  by  turning  the  lamina  round  in  its  own  plane. 

Following  gradually  in  this  manner  the  direction  of  the  polarisation 
given  to  a  molecule  of  light,  transmitted  through  different  thicknesses 
of  a  crystalline  medium,  it  will  be  found  to  undergo  periodical  alter- 
nations, which,  if  light  be  a  matter  emitted,  indicate  an  oscillatory 
motion  of  the  axes  of  the  molecules  accompanying  their  progressive 
motion.  M.  Biot  has  designated  the  fact  by  the  name  of  moveable 
polarisation,  which  is  merely  the  expression  of  results  observed. 

If  the  system  of  undulations  be  adopted,  the  colours  of  the  two 
images  may  be  attributed  to  the  interference  of  the  two  pencils  into 
which  the  incident  polarised  light  separates  in  passing  through  the 
lamina.  This  is  what  Dr  Young  does,  and  it  is  remarkable  that 
calculations  founded  on  this  principle  gave  him  the  nature  of  the 


344  Optics. 

tints,  and  the  periods  after  which  they  recur,  precisely  as  M.  Biot 
had  determined  them  by  experiment.  As  to  the  alternations  of  po- 
larisation, they  become,  in  the  undulation  system,  a  compound  result 
produced  by  the  mutual  influence  of  the  interfering  rays,  and  it  is 
easy  to  deduce  from  observation  the  conditions  to  which  the  mixture 
of  the  waves  must  be  subjected  to  produce  the  new  direction  of  ap- 
parent polarisation.  M.  Fresnel  has  done  this,  and  the  indications 
of  his  formula  have  been  found  conformable  in  all  respects  to  the 
laws  deduced  by  M.  Biot  from  observation. 

These  interferences  of  the  rays  may  be  produced  without  the 
assistance  of  crystalline  laminae  ;  we  may  equally  employ  thick  plates, 
provided  the  rays  pass  through  them  at  very  small  inclinations  to 
their  crystalline  axes.  If  the  experiment  be  made  with  a  conical 
pencil  of  light,  large  enough  to  give  the  various  rays  composing  its 
inclinations  sensibly  different  to  the  axes,  so  that  they  experience 
double  refractions, sensibly  unequal,  these  rays,  analyzed  after  they 
emerge,  offer  different  colours  united  in  the  same  system  of  polarisa- 
tion ;  and  the  union  of  these  colours  forms  round  the  axes  coloured 
zones,  the  configuration  of  which  indicates  the  system  of  polarising 
action  exerted  by  the  substance  under  consideration.  This  kind  of 
experiment  is,  therefore,  very  proper  to  exhibit  the  axes  and  to  indi- 
cate the  mode  of  polarisation  with  which  any  given  substance  affects 
the  rays. 

Upon  the  whole,  the  interferences  of  polarised  rays  offer  very 
remarkable  properties,  many  of  which  have  been  discovered  aiul 
analyzed  by  MM.  Arago  and  Fresnel,  with  great  ingenuity  and  con- 
siderable success,  but  as  the  limits  of  this  work  do  not  allow  of  a  full 
exposition  of  them,  I  will  only  cite  one,  which  is,  that  rays  polarised 
at  right  angles  do  not  affect  each  other  when  they  are  made  to  inter- 
fere, whereas  they  preserve  that  power  when  they  are  polarised  in 
the  same  direction.  It  is  not  only  crystalline  bodies  that  modify 
polarisation  impressed  on  the  rays  of  light ;  MM.  Mains  and  Biot 
found  by  different  experiments  made  about  the  same  time,  that  if  a 
ray  be  refracted  successively  by  several  glass  plates  placed  parallel 
to  each  other,  it  will  at  length  be  polarised  in  one  single  direction 
perpendicular  to  the  plane  of  refraction.  Malus,  by  a  very  ingenious 
analysis  of  this  phenomenon,  has  moreover  shown  that  it  is  progres- 
sive, the  first  glass  polarising  a  small  portion  of  the  incident  light,  the 
second  a  part  of  that  which  had  escaped  the  action  of  the  first,  and 
so  on.  M.  Arago,  measuring  the  successive  intensities  by  a  method 


Polarisation  of  Light.  345 

of  his  own  invention,  has  shown  that  they  are  exactly  equal  to  the 
quantity  of  light  polarised  in  contrary  directions  at  each  reflection.  A 
phenomenon  analogous  to  this  is  produced  naturally  in  prisms  of 
tourmaline,  which  appear  to  be  composed  of  a  multitude  of  smaller 
prisms,  united  together,  but  without  any  immediate  contact.  All 
light  passing  through  one  of  these  prisms  perpendicularly,  is  found  to 
be  polarised  in  a  direction  perpendicular  to  the  edges,  so  that  if  two 
such  prisms  be  placed  at  right  angles,  on  looking  through  them  a 
dark  spot  is  seen  where  they  cross.  This  property  of  the  tourma- 
line affords  a  very  convenient  method  to  impress  on  a  pencil  of  rays 
a  polarisation  in  any  required  direction,  or  to  discover  such  polarisa- 
tion when  it  exists. 

Moreover,  M.  Biot  has  discovered  that  certain  solid  bodies,  and 
even  certain  fluids,  possess  the  faculty  of  changing  progressively  the 
polarisation  previously  impressed  on  rays  passing  through  them  ;  and, 
by  an  analysis  of  the  phenomena  produced  by  those  substances,  he 
has  shown  that  the  same  faculty  resides  in  their  smallest  molecules, 
so  that  they  preserve  it  in  all  states,  solid,  liquid,  and  aeriform, 
and  even  in  all  combinations  into  which  they  may  happen  to  enter. 
M.  Fresnel  has  found  certain  analogies  between  these  phenomena  and 
those  of  double  refraction,  which  seem  to  connect  the  two  together 
most  intimately  through  the  mediation  of  total  reflection. 

Since  reflection  and  refraction,  even  of  the  ordinary  kind,  modify 
the  polarisation  of  light,  we  may  expect  to  find  this  effect  produced 
when  rays  of  light  are  made  to  pass  through  media  of  regularly  vary- 
ing density.  It  is  accordingly  found  that  all  transparent  bodies  which 
are  sufficiently  elastic  to  admit  of  different  positions  of  their  particles 
round  a  given  state  of  equilibrium,  as  glass,  crystals,  animal  jellies, 
horn,  &c.  produce  phenomena  of  polarisation  when  they  are  com- 
pressed or  expanded,  or  made  unequally  dense  by  being  considera- 
bly heated,  and  then  cooled  suddenly  and  unequally.  These  phe- 
nomena, discovered  originally  by  Dr  Seebeck,  have  been  since  studi- 
ed and  considerably  extended  by  Dr  Brewster,  who  has  moreover 
remarked,  that  successive  reflections  of  light  on  metallic  plates  pro- 
duced phenomena  of  colours  in  which  both  M.  Biot  and  he  have 
recognised  all  the  characters  of  alternate  polarisation. 

Knowing,  by  what  has  preceded,  the  experimental  laws,  accord- 
ing to  which  light  is  decomposed  in  crystals  endued  with  double  re- 
fraction, we  may  consider  these  effects  as  proofs  proper  to  charac- 
terize the  mode  of  intimate  aggregation  of  the  particles  of  such 

Elem.  44 


346  Optics. 

bodies,  and  to  give  some  insight  into  the  nature  of  their  crystalline 
structure.  Light  becomes  thus,  as  i  were,  a  delicate  sounding  in- 
strument with  which  we  probe  the  substance  of  matter,  and  which, 
insinuating  itself  between  their  minutest  parts,  permits  us  to  study 
their  arrangement,  at  which  mineralogists  previously  guessed  only  by 
inspection  of  their  external  forms.  M.  Biot  has  shown  the  use  of 
this  method,  applying  it  to  a  numerous  class  of  minerals  designated 
by  the  general  name  of  mica,  and  he  thinks  he  has  decisive  reasons 
to  believe  that  several  substances  of  natures  extremely  different 
as  to  their  composition  and  structure  have  been  improperly  compris- 
ed under  that  name.  He  has  also  made  use  of  the  phenomena  of 
alternate  polarisation,  to  construct  an  instrument  which  he  calls  a 
colorigrade,  which,  producing  in  all  cases  the  same  series  of  colours 
in  exactly  the  same  order,  merely  by  the  nature  of  its  construction, 
affords  a  mode  of  designation  just  as  convenient  for  comparison  as 
that  furnished  by  the  thermometer  for  temperatures. 

Many  other  experiments  have  been  made,  and  are  daily  making ; 
many  other  properties  have  been  discovered  in  polarised  light ;  but 
the  limits  of  this  work  do  not  allow  us  to  give  any  detailed  account 
of  them,  so  that  we  have  been  obliged  to  confine  ourselves  to  the 
results,  which  are,  perhaps  not  the  most  important  part  of  the  sub- 
ject, but  the  easiest  to  explain  ;  our  aim  in  this  rapid  sketch  being 
rather  to  stimulate  than  satisfy  the  desire  of  knowledge  on  this  branch 
of  science,  which  presents  so  vast  a  Geld  for  research  both  in  theory 
and  experiment,  and  which,  though  so  lately  discovered,  has  already 
furnished  some  useful  applications  to  physics  and  mineralogy. 


TABLE 


Of  the  Refractive  and  Dispersive  Powers  of  different  Substances, 
with  their  Densities  compared  with  that  of  Water,  which  is 
taken  as  the  Unit. 

The  substances  marked  (*)  are  combustible. 
The  refraction  is  supposed  to  take  place  between  the  given  substance  and  a  vacuum. 


Substances. 

Ratio  of 
refraction. 

Dispersive 
po\\  er. 

Density. 

Chromate  of  lead  (strongest) 

2,974 
2  549 

0,4 
0  267 

5,8 

Q    A 

Chromate  of  lead  (weakest) 
*Diamoud        
*Sulphur  (native)     .     .     .     . 
Carbonate  of  lead  (strongest) 
weakest       

2,503 
2,45 
2,115 
2,084 
1,813 
1,815 

0,262 
0,038 

|  0,091 
0033 

0,4 

5,8 
3,521 
2,033 
6,071 
4,000 
3  Q13 

1,735 

0,030 

Calcareous  Spar  (strongest) 

1,665 
1,519 

0,04 

}  2,715 

*Oil  of  Cassia  1     

1,641 

0,139 

Flint  glass  ....          . 

1,616 

0,048 

3,329 

another  kind    . 

1,590 
1,562 

0,026 

2,653 

Rock  salt    .*    . 

1,557 

0,053 

2,130 

Canada  balsam    

1,549 
1,544 

0,045 
0,036 

2,642 

1,536 

0,037 

2,322 

Plate  glass 

1,527 

0,032 

2,488 

1,512 

0,036 

1,452 

*Oil  of  almonds  
*Oil  of  turpentine     .     .     .     . 

1,483 
1,475 
1,475 

0,042 
0,030 

0,917 
0,869 
1,718 

Sulphuric  acid     

1,440 
1,436 

0,031 
0,022 

1,850 
3,168 

1,406 

0,045 

1.217 

Muriatic  acid  
*Alcohol                    .     .     . 

1,374 
1,374 

0,043 
0,029 

1,194 
0,825 

1,361 

0,037 

1,090 

1,343 

1,026 

Water                        .     .     • 

1,336 

0,035 

1,000 

Ice         

1,307 

0,930 

Air    .     •     
Oxygen       ...... 

1,00029 
1,00028 
1,00014 

0,0013 
0,0014 
0,0001 

1,00029 

0,0012 

Carbonic  acid  gas     .     .     • 

1,00045 

0,0018 

THB  END. 


